\section{Introduction}\label{SEC:Introduction}
\subsection{Tur\'{a}n number and stability}\label{SUBSEC:turan-number-and-stability}
For $r\ge 2$ an $r$-uniform hypergraph (henceforth $r$-graph) $\mathcal{H}$ is a collection of $r$-subsets of some finite set $V$.
Given a family $\mathcal{F}$ of $r$-graphs we say $\mathcal{H}$ is $\mathcal{F}$-free
if it does not contain any member of $\mathcal{F}$ as a subgraph.
The {\em Tur\'{a}n number} $\mathrm{ex}(n,\mathcal{F})$ of $\mathcal{F}$ is the maximum
number of edges in an $\mathcal{F}$-free $r$-graph on $n$ vertices.
The {\em Tur\'{a}n density} $\pi(\mathcal{F} )$ of $\mathcal{F}$ is defined as
$\pi(\mathcal{F})=\lim_{n\to \infty}\mathrm{ex}(n,\mathcal{F})/{n\choose r}$.
A family $\mathcal{F}$ is called \emph{nondegenerate} if $\pi(\mathcal{F}) >0$.
The study of $\mathrm{ex}(n,\mathcal{F})$ is perhaps the central topic in extremal graph and hypergraph theory.

Much is known about $\mathrm{ex}(n,\mathcal{F})$ when $r=2$,
and one of the most famous results in this regard is Tur\'{a}n's theorem,
which states that for every integer $\ell \ge 2$ the Tur\'{a}n number $\mathrm{ex}(n,K_{\ell+1})$
is uniquely achieved by the balanced $\ell$-partite graph on $n$ vertices,
which is called the Tur\'{a}n graph $T(n,\ell)$.

For $r\ge 3$ determining $\pi(\mathcal{F})$ for a family $\mathcal{F}$ of $r$-graphs
is known to be notoriously hard in general.
Indeed, the problem of determining $\pi(K_{\ell}^{r})$ raised by Tur\'{a}n~\cite{TU41},
where $K_{\ell}^{r}$ is the complete $r$-graph on $\ell$ vertices, is still wide open for all $\ell>r\ge 3$.
Erd\H{o}s offered $\$ 500$ for the determination of any $\pi(K_{\ell}^{r})$
with $\ell > r \ge 3$ and $\$ 1000$ for the determination of all $\pi(K_{\ell}^{r})$ with $\ell > r \ge 3$.

\begin{conjecture}[Tur\'{a}n~\cite{TU41}]\label{CONJ:Turan-complete-r-graph}
For every integer $\ell\ge 3$ we have $\pi(K_{\ell+1}^{3}) = 1- {4}/{\ell^2}$.
\end{conjecture}

The case $\ell = 3$ above, which states that $\pi(K_{4}^{3}) = 5/9$ has generated a lot of interest and activity over the years.
Many constructions (e.g. see \cite{BR83,KO82,FO88}) are known to achieve the value in Conjecture~\ref{CONJ:Turan-complete-r-graph}
for $\ell = 3$.
In particular, Kostochka~\cite{KO82} showed that there are at least $2^{n-2}$ nonisomorphic
extremal $K_{4}^{3}$-free constructions on $3n$ vertices (assuming Tur\'{a}n's Tetrahedron conjecture is true).
This is perhaps one of the reasons why it is so challenging.
On the other hand, successively better upper bounds for $\pi(K_{4}^{3})$ were obtained by
de Caen $\cite{DC88}$, Giraud (see $\cite{CL99}$), Chung and Lu  $\cite{CL99}$, and Razborov $\cite{RA10}$.
The current record is $\pi(K_{4}^{3}) \le 0.561666$, which was obtained by Razborov  $\cite{RA10}$ using the Flag Algebra machinery.

In this work, we continue the research on hypergraph Tur\'{a}n problems with the flavour
of trying to show that certain phenomena impossible for graphs are possible for $r$-graphs when $r\ge 3$.
There are several results that belong to this category.
For instance, the well known work of Frankl and R\"odl on jumps~\cite{FR84},
and Pikhurko's theorem that for $r\ge 3$ there exist uncountably many Tur\'an densities of infinite families of $r$-graphs~\cite{PI14}.
More specifically, we will show that there are infinitely many finite families $\mathcal{F}$ with similar properties
as $K_{4}^{3}$ (assuming Tur\'an's conjecture) in terms of the number of extremal constructions.

Many families $\mathcal{F}$ have the property that there is a unique $\mathcal{F}$-free hypergraph
$\mathcal{G}$ on $n$ vertices achieving $\mathrm{ex}(n,\mathcal{F})$,
and moreover, any $\mathcal{F}$-free hypergraph $\mathcal{H}$ of size close to $\mathrm{ex}(n,\mathcal{F})$
can be transformed to $\mathcal{G}$ by deleting and adding very few edges.
Such a property is called the stability of $\mathcal{F}$.
The first stability theorem, which says that $K_{\ell}$ is stable for all integers $\ell\ge 3$,
was proved independently by Erd\H{o}s and Simonovits $\cite{SI68}$.
Their result motivated the third author~\cite{MU07} to make the following definition.

\begin{definition}[$t$-stable]\label{DFN:t-stable}
Let $r \ge 2$ and $t \ge 1$ be integers.
A family $\mathcal{F}$ of $r$-graphs is $t$-stable if for every $m\in\mathbb{N}$
there exist $r$-graphs $\mathcal{G}_{1}(m),\ldots,\mathcal{G}_{t}(m)$ on $m$ vertices such that the following holds.
For every $\delta >0$ there exist $\epsilon > 0$ and $N_0$ such that for all $n \ge N_0$
if $\mathcal{H}$ is an $\mathcal{F}$-free $r$-graph on $n$ vertices with $|\mathcal{H}| > (1-\epsilon) \mathrm{ex}(n,\mathcal{F})$
then $\mathcal{H}$ can be transformed to some $\mathcal{G}_{i}(n)$ by adding and removing at most $\delta n^r$ edges.
Say $\mathcal{F}$ is stable if it is $1$-stable.
\end{definition}

Denote by $\xi(\mathcal{F})$ the minimum integer $t$ such that $\mathcal{F}$ is $t$-stable,
and set $\xi(\mathcal{F}) = \infty$ if there is no such $t$.
Call $\xi(\mathcal{F})$ \emph{the stability number} of $\mathcal{F}$.

The classical Erd\H{o}s--Stone--Simonovits theorem~\cite{ES46,ES66} and Erd\H{o}s--Simonovits stability theorem~\cite{SI68}
imply that every nondegenerate family of graphs is stable.
For hypergraphs there are many families (whose Tur\'{a}n densities are unknown)
which are conjecturally not stable.
Two famous examples are Tur\'{a}n's conjecture on tetrahedra (i.e. $K_{4}^3$) mentioned above (e.g. see~\cite{AS95,KO82,LM22}),
and the Erd\H{o}s--S\'{o}s conjecture on triple systems with bipartite links (e.g. see~\cite{FF84,LM22}).
In fact, families that are not stable and whose Tur\'{a}n densities can be determined were constructed only very recently.
In~\cite{LM22}, the third and fourth authors constructed the first finite $2$-stable family of $3$-graphs.
Later in~\cite{LMR1}, together with Reiher, they further constructed the first finite $t$-stable family of $3$-graphs
for every integer $t\ge 3$.
In~\cite{LMR3}, using a similar method that was used for the hypergraph jump problem (see e.g.~\cite{FPRT07})
their constructions were extended to every integer $r \ge 4$.

Using Kostochka's constructions the authors proved in~\cite{LM22} that $\xi(K_{4}^3) = \infty$ (assuming Tur\'{a}n's conjecture).
However, the methods used in~\cite{LM22} and~\cite{LMR1}
for constructing the $2$-stable family and $t$-stable families cannot be extended further to
construct a finite family whose stability number is infinite, and
this was left as an open question in both papers.

Our main result in this work is to give a novel method to construct (finite) families $\mathcal{F}$ with $\xi(\mathcal{F}) = \infty$.
Our method is based on a simple property of multilinear polynomials,
which will be stated in Section~\ref{PROP:multilinear-polynomial}.
We will use this method to give infinitely many (finite) families of $3$-graphs
whose number of extremal constructions grows with the number of vertices.
Further, using the machinery provided in~\cite{LMR2} we also prove an Andr\'{a}sfai--Erd\H{o}s--S\'{o}s type~\cite{AES74}
stability theorem for these families.


We identify an $r$-graph $\mathcal{H}$ with its edge set,
use $V(\mathcal{H})$ to denote its vertex set, and denote by $v(\mathcal{H})$ the size of $V(\mathcal{H})$.
An $r$-graph $\mathcal{H}$ is a \emph{blowup} of an $r$-graph $\mathcal{G}$ if there exists a
map $\psi\colon V(\mathcal{H}) \to V(\mathcal{G})$ so that $\psi(E) \in \mathcal{G}$ iff $E\in \mathcal{H}$.
We say $\mathcal{H}$ is \emph{$\mathcal{G}$-colorable}
if there exists a map $\phi\colon V(\mathcal{H}) \to V(\mathcal{G})$ so that $\phi(E)\in \mathcal{G}$ for all $E\in \mathcal{H}$,
and we call such a map $\phi$ a \emph{homomorphism} from $\mathcal{H}$ to $\mathcal{G}$.
In other words, $\mathcal{H}$ is $\mathcal{G}$-colorable if and only if $\mathcal{H}$ occurs as a subgraph in some blowup
of $\mathcal{G}$.

\begin{theorem}\label{THM:main-sec1.1}
For every integer $t\ge 3$ there exists a finite family $\mathcal{F}_t$ of $3$-graphs such that the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item We have $\mathrm{ex}(n,\mathcal{F}_t) \le \frac{(t-2)(t-1)}{6t^2}n^3$ for all $n\in \mathbb{N}$,
        and equality holds whenever $t \mid n$.
\item If $t \mid n$, then the number of nonisomorphic maximum $\mathcal{F}_t$-free $3$-graphs on $n$ vertices is
        at least ${n}/{2t}$.
\item We have $\xi(\mathcal{F}_t) = \infty$.
\item For every integer $t\ge 4$ there exist constants $\epsilon = \epsilon(t)>0$
        and $N_0 = N_0(t)$ such that the following holds for every integer $n\ge N_0$.
        Every $n$-vertex $\mathcal{F}_t$-free $3$-graph with minimum degree at least $\left(\frac{(t-2)(t-1)}{2t^2}-\epsilon\right)n^2$
        is $\Gamma_{t}$-colorable, where $\Gamma_{t}$ is some fixed $3$-graph on $t+2$ vertices.
\end{enumerate}
\end{theorem}

Motivated by the hypergraph jump problem\footnote{Our results in this paper are not very related to the Jump problem, so we omit the related definitions here.
We refer the interested reader to e.g. \cite{FR84,FPRT07,BT11} for related results.}, the authors asked in~\cite{LMR1}
whether there exists a finite family whose Tur\'{a}n density is $2/9$ that is not stable?
By letting $t = 3$ in Theorem~\ref{THM:main-sec1.1} we see that the finite family $\mathcal{F}_3$ of $3$-graphs
has Tur\'{a}n density $2/9$ but $\xi(\mathcal{F}_3) = \infty$, thus answering this question.
\subsection{Multilinear polynomials}\label{SEC:Multilinear-polynomials}
In this short section, we state a simple result about polynomials. Later we will see that our main theorem reduces to this result.

Denote by $\Delta_{m-1}$ the standard $(m-1)$-dimensional simplex,
i.e.
\begin{align}
\Delta_{m-1} = \left\{(x_1,\ldots,x_m)\in [0,1]^m\colon x_1+\cdots+x_m= 1\right\}. \notag
\end{align}
Given an $m$-variable continuous function $f$ we define
\begin{align}
\lambda(f) = \max\left\{f(x_1,\ldots,x_m)\colon (x_1,\ldots,x_m)\in \Delta_{m-1}\right\},  \notag
\end{align}
and
\begin{align}
Z(f) = \left\{(x_1,\ldots,x_m)\in \Delta_{m-1}\colon f(x_1,\ldots,x_m)-\lambda(f) = 0\right\}.  \notag
\end{align}
Since $\Delta_{m-1}$ is compact,
a well known theorem of Weierstra\ss\ implies that the restriction of $f$ to $\Delta_{m-1}$
attains a maximum value. Thus $\lambda(f)$ and $Z(f)$ are well-defined.

Let $p(X_1,\ldots,X_m)$ be a polynomial, where $X_1,\ldots,X_m$ are indeterminates.
We say $p$ is \emph{multilinear} if for every $\alpha \in \mathbb{R}$ and for every $i\in [m]$
we have $p(X_1,\ldots,\alpha \cdot X_i ,\ldots,X_m) = \alpha \cdot p(X_1,\ldots, X_i ,\ldots,X_m)$. Note that this is equivalent to the fact that each term of $p$ is of the form $\prod_{i \in S} x_i$ for some $S$.
We say $p$ is \emph{nonnegative} (or \emph{nonpositive}) if $p(x_1,\ldots,x_m)\ge 0$ (or $p(x_1,\ldots,x_m)\le 0$)
for all $(x_1,\ldots,x_m)\in \Delta_{m-1}$.
For a pair $\{i,j\}\subset [m]$ we say $p$ is symmetric with respect to $X_i$ and $X_j$ if
$$p(X_1,\ldots,X_i,\ldots, X_j, \ldots, X_m)=p(X_1,\ldots,X_j,\ldots, X_i, \ldots, X_m).$$

An easy observation is that a multilinear polynomial $p(X_1,\ldots,X_m)$ is symmetric with respect to $X_i$ and $X_j$
iff there exist polynomials $p_1, p_2, p_3$ without indeterminates $X_i$ and $X_j$ such that
$p = p_1 + p_2 (X_i + X_j) + p_3 X_iX_j$.

Given two vectors $\vec{x}, \vec{y}\in \mathbb{R}^m$ define the line segment $L(\vec{x}, \vec{y})$
with endpoints $\vec{x}$ and $\vec{y}$ as
\begin{align}
L(\vec{x}, \vec{y}) = \left\{\alpha\cdot \vec{x}+(1-\alpha)\cdot \vec{y}\colon \alpha \in [0,1]\right\}. \notag
\end{align}

We will later prove the following result about multilinear polynomials.

\begin{proposition}\label{PROP:multilinear-polynomial}
Let $p(X_1,\ldots,X_m) = p_1 + p_2 (X_i + X_j) + p_3 X_iX_j$
be an $m$-variable multilinear polynomial that is symmetric with respect to $X_i$ and $X_j$.
Suppose that $p_3$ is nonnegative,
and $p_4, p_5$ are nonnegative polynomials satisfying $p_4+p_5=p_3$.
Then the $(m+2)$-variable polynomial
\begin{align}
& \hat{p}(X_1,X'_1,X_2,X'_2,X_3,\ldots,X_m)  \notag\\
&  = p_1 + p_2(X_1+X'_1+X_2+X'_2)
            + p_{4}(X_1+X'_1)(X_2+X'_2)
            + p_{5}(X_1+X_2)(X'_1+X'_2) \notag
\end{align}
satisfies $\lambda(\hat{p}) = \lambda(p)$, and moreover,
for every $(x_1,\ldots, x_m)\in Z(p)$ we have $L(\vec{y}, \vec{z}) \subset Z(\hat{p})$,
where $\vec{y}, \vec{z}\in \Delta_{m+1}$ are defined by
\begin{align}
\vec{y}
&  = \left(x_1,\ldots,x_{i-1},(x_i+x_j)/2, 0,x_{i+1}, \ldots,x_{j-1}, 0,(x_i+x_j)/2, x_{j+1}, \ldots,  x_m\right), \notag\\
\vec{z}
&  = \left(x_1,\ldots,x_{i-1},0, (x_i+x_j)/2,x_{i+1}, \ldots,x_{j-1},(x_i+x_j)/2, 0,x_{j+1}, \ldots,  x_m\right). \notag
\end{align}
\end{proposition}


\subsection{Nonminimal hypergraphs}\label{SUBSEC:nonminimal-hypergraphs}
Now we return to hypergraphs.
Our constructions are closely related to the so-called nonminimal hypergraphs.
To define them properly we need some definitions related to the Lagrangian of a hypergraph,
which was introduced by Frankl and R\"odl in~\cite{FR84}.

For an $r$-graph $\mathcal{G}$ on $m$ vertices
(let us assume for notational transparency that $V(\mathcal{G}) = [m]$) the multilinear polynomial
$p_{\mathcal{G}}$ is defined by
\begin{align}
p_{\mathcal{G}}(X_1,\ldots,X_m) = \sum_{E\in \mathcal{G}}\prod_{i\in E}X_i. \notag
\end{align}
The \emph{Lagrangian} of $\mathcal{G}$ is defined by $\lambda(\mathcal{G}) = \lambda(p_{\mathcal{G}})$.
Define
\begin{align}
Z(\mathcal{G}) = Z(p_{\mathcal{G}})
= \left\{(x_1,\ldots,x_n)\in \Delta_{m-1}\colon p_{\mathcal{G}}(x_1,\ldots,x_m) = \lambda(\mathcal{G})\right\}. \notag
\end{align}

For a vertex set $S\subset V(\mathcal{G})$ we use
$\mathcal{G}-S$ to denote the induced subgraph of $\mathcal{G}$ on $V(\mathcal{G})\setminus S$.

\begin{definition}
An $r$-graph $\mathcal{G}$ is minimal if $\lambda(\mathcal{G}-v)< \lambda(\mathcal{G})$
for every vertex $v\in \mathcal{G}$. Otherwise, it is nonminimal.
\end{definition}

A substantial amount of research on hypergraph Tur\'{a}n problems is about minimal hypergraphs.
Indeed,  for every nondegenerate family $\mathcal{F}$ of hypergraphs that was studied before this work,
the extremal $\mathcal{F}$-free constructions are either minimal or blowups of minimal hypergraphs or close to them in edit-distance.
Conversely, in~\cite{PI14}, Pikhurko proved that for every minimal hypergraph $\mathcal{G}$
there exists a finite family $\mathcal{F}$ whose unique extremal construction is a blowup of $\mathcal{G}$
\footnote{Pikhurko's results are much more general than what we stated here,
and we refer the reader to~\cite{PI14} for more details.}.
In contrast, there was very little literature about nonminimal hypergraphs.

Parallel to Plikurko's result about minimal hypergraphs we have the following result about nonminimal hypergraphs.

\begin{proposition}\label{PROP:nonminal-finite-family}
Let $r\ge 3$ be an integer.
For every nonminial $r$-graph $\mathcal{G}$ there exists a finite family $\mathcal{F}$ of $r$-graphs
such that for every integer $n\in\mathbb{N}$
there exists a maximum $\mathcal{F}$-free $r$-graph on $n$-vertices which is a blowup of $\mathcal{G}$.
\end{proposition}


Next, we introduce an operation motivated by Proposition~\ref{PROP:multilinear-polynomial}.
It shows  that for a certain class of $3$-graphs $\mathcal{G}$
we can produce a nonminimal $3$-graph containing $\mathcal{G}$.

For a $3$-graph $\mathcal{G}$ and a pair of vertices $\{u,v\}\subset V(\mathcal{G})$
the \emph{neighborhood} $N_{\mathcal{G}}(u,v)$ and \emph{codegree} $d_{\mathcal{G}}(u,v)$ of $\{u,v\}$ are defined as
$$N_{\mathcal{G}}(u,v) = \{w\in\mathcal{G}\colon \{u,v,w\}\in\mathcal{G}\}
\quad\text{and}\quad d_{\mathcal{G}}(u,v) = |N_{\mathcal{G}}(u,v)|.$$
We say $\mathcal{G}$ is \emph{$2$-covered} if $d_{\mathcal{G}}(u,v) \ge 1$ for every pair $\{u,v\}\subset V(\mathcal{G})$. For a vertex $u \in V(\mathcal{G})$
the \emph{link} of $v$ is defined as
$$L_{\mathcal{G}}(u) = \{\{v,w\}\subset\mathcal{G}\colon \{u,v,w\}\in\mathcal{G}\}.$$ Vertices $u$ and $u'$ in $\mathcal{G}$ are clones if
$L_{\mathcal{G}}(u)=L_{\mathcal{G}}(v)$.

\begin{definition}[Crossed blowup]
Let $\mathcal{G}$ be a $3$-graph and
$\{v_1,v_2\}\subset \mathcal{G}$ be a pair of vertices with $d(v_1,v_2) = k \ge 2$.
Fix an ordering of vertices in $N_{\mathcal{G}}(v_1,v_2)$, say $N_{\mathcal{G}}(v_1,v_2) = \{u_1,\ldots, u_k\}$.
The crossed blowup $\mathcal{G}\boxplus\{v_1,v_2\}$ of $\mathcal{G}$ on $\{v_1,v_2\}$ is defined as follows.
\begin{enumerate}[label=(\alph*)]
\item Remove all edges containing the pair $\{v_1,v_2\}$ from $\mathcal{G}$,
\item add two new vertices $v_1'$ and $v_2'$, make $v_1'$ a clone of $v_1$ and $v_2'$ a clone of $v_2$,
\item for every $i\in [k-1]$ add the edge set $\{u_iv_1v_2,u_iv_1v_2',u_iv_1'v_2,u_iv_1'v_2'\}$,
                and for $i = k$ add the edge set $\{u_kv_1v_1',u_kv_1v_2',u_kv_2v_1',u_kv_2v_2'\}$.
\end{enumerate}
\end{definition}

\textbf{Remark.}
If in Step (c) we add the edge set $\{u_kv_1v_2,u_kv_1v_2',u_kv_1'v_2,u_kv_1'v_2'\}$
instead of $\{u_kv_1v_1',u_kv_1v_2',u_kv_2v_1',u_kv_2v_2'\}$,
then we obtain an ordinary blowup of $\mathcal{G}$.


\begin{figure}[htbp]
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (164,341.57) -- (229,341.6) -- (255,216.57) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (164,341.57) -- (229,341.6) -- (198,216.57) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (164,341.57) -- (229,341.6) -- (141,216.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (137,216.57) .. controls (137,214.36) and (138.79,212.57) .. (141,212.57) .. controls (143.21,212.57) and (145,214.36) .. (145,216.57) .. controls (145,218.78) and (143.21,220.57) .. (141,220.57) .. controls (138.79,220.57) and (137,218.78) .. (137,216.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (194,216.57) .. controls (194,214.36) and (195.79,212.57) .. (198,212.57) .. controls (200.21,212.57) and (202,214.36) .. (202,216.57) .. controls (202,218.78) and (200.21,220.57) .. (198,220.57) .. controls (195.79,220.57) and (194,218.78) .. (194,216.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (251,216.57) .. controls (251,214.36) and (252.79,212.57) .. (255,212.57) .. controls (257.21,212.57) and (259,214.36) .. (259,216.57) .. controls (259,218.78) and (257.21,220.57) .. (255,220.57) .. controls (252.79,220.57) and (251,218.78) .. (251,216.57) -- cycle ;
\draw  [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 }  ,fill opacity=1 ] (402,206.57) .. controls (402,204.36) and (403.79,202.57) .. (406,202.57) .. controls (408.21,202.57) and (410,204.36) .. (410,206.57) .. controls (410,208.78) and (408.21,210.57) .. (406,210.57) .. controls (403.79,210.57) and (402,208.78) .. (402,206.57) -- cycle ;
\draw  [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 }  ,fill opacity=1 ] (466,206.57) .. controls (466,204.36) and (467.79,202.57) .. (470,202.57) .. controls (472.21,202.57) and (474,204.36) .. (474,206.57) .. controls (474,208.78) and (472.21,210.57) .. (470,210.57) .. controls (467.79,210.57) and (466,208.78) .. (466,206.57) -- cycle ;
\draw  [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][fill={rgb, 255:red, 248; green, 231; blue, 28 }  ,fill opacity=1 ] (529,206.57) .. controls (529,204.36) and (530.79,202.57) .. (533,202.57) .. controls (535.21,202.57) and (537,204.36) .. (537,206.57) .. controls (537,208.78) and (535.21,210.57) .. (533,210.57) .. controls (530.79,210.57) and (529,208.78) .. (529,206.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (160,341.57) .. controls (160,339.36) and (161.79,337.57) .. (164,337.57) .. controls (166.21,337.57) and (168,339.36) .. (168,341.57) .. controls (168,343.78) and (166.21,345.57) .. (164,345.57) .. controls (161.79,345.57) and (160,343.78) .. (160,341.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (225,341.6) .. controls (225,339.39) and (226.79,337.6) .. (229,337.6) .. controls (231.21,337.6) and (233,339.39) .. (233,341.6) .. controls (233,343.81) and (231.21,345.6) .. (229,345.6) .. controls (226.79,345.6) and (225,343.81) .. (225,341.6) -- cycle ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (432,281.57) -- (501,350.57) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (436,277.57) -- (505,346.57) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (432,346.57) -- (501,277.57) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (436,350.57) -- (505,281.57) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (432,277.57) -- (505,277.57) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (432,350.57) -- (505,350.57) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (505,277.57) -- (505,350.57) ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (501,277.57) .. controls (501,275.36) and (502.79,273.57) .. (505,273.57) .. controls (507.21,273.57) and (509,275.36) .. (509,277.57) .. controls (509,279.78) and (507.21,281.57) .. (505,281.57) .. controls (502.79,281.57) and (501,279.78) .. (501,277.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (501,350.57) .. controls (501,348.36) and (502.79,346.57) .. (505,346.57) .. controls (507.21,346.57) and (509,348.36) .. (509,350.57) .. controls (509,352.78) and (507.21,354.57) .. (505,354.57) .. controls (502.79,354.57) and (501,352.78) .. (501,350.57) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (432,277.57) -- (432,350.57) ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (428,350.57) .. controls (428,348.36) and (429.79,346.57) .. (432,346.57) .. controls (434.21,346.57) and (436,348.36) .. (436,350.57) .. controls (436,352.78) and (434.21,354.57) .. (432,354.57) .. controls (429.79,354.57) and (428,352.78) .. (428,350.57) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (428,277.57) .. controls (428,275.36) and (429.79,273.57) .. (432,273.57) .. controls (434.21,273.57) and (436,275.36) .. (436,277.57) .. controls (436,279.78) and (434.21,281.57) .. (432,281.57) .. controls (429.79,281.57) and (428,279.78) .. (428,277.57) -- cycle ;
\draw   (317,295.37) -- (331.19,295.37) -- (331.19,292.37) -- (340.66,298.37) -- (331.19,304.37) -- (331.19,301.37) -- (317,301.37) -- cycle ;
\draw (128,195) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_1$};
\draw (187,195) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_2$};
\draw (245,195) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_3$};
\draw (156,355) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v_1$};
\draw (220,355) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v_2$};
\draw (398,182) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_1$};
\draw (461,182) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_2$};
\draw (525,182) node [anchor=north west][inner sep=0.75pt]   [align=left] {$u_3$};
\draw (410,272) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v_1$};
\draw (515,272) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v_2$};
\draw (410,350) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v'_1$};
\draw (515,350) node [anchor=north west][inner sep=0.75pt]   [align=left] {$v'_2$};
\end{tikzpicture}
\caption{$\{u_1v_1v_2,u_2v_1v_2,u_3v_1v_2\}$ and $\{u_1v_1v_2,u_2v_1v_2,u_3v_1v_2\}\boxplus\{v_1,v_2\}$.
        The link of red vertices is the red $K_{2,2}$, the link of the yellow vertex is the yellow $K_{2,2}$.}
\end{figure}

\begin{figure}[htbp]
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ][line width=1.5]    (197.5,166.5) -- (133,209) -- (197.67,93) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.9 ][line width=1.5]    (197.67,93) -- (197.5,166.5) -- (262.13,209) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.7 ][line width=1.5]    (197.5,166.5) -- (262.13,209) -- (133,209) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (456,50) -- (456,94) -- (416,145) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (456,50) -- (456,94) -- (489,145) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (456,50) -- (456,94) -- (416,221) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.5 ]   (456,50) -- (456,94) -- (489,221) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (194,166.5) .. controls (194,164.57) and (195.57,163) .. (197.5,163) .. controls (199.43,163) and (201,164.57) .. (201,166.5) .. controls (201,168.43) and (199.43,170) .. (197.5,170) .. controls (195.57,170) and (194,168.43) .. (194,166.5) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (194.17,93) .. controls (194.17,91.07) and (195.74,89.5) .. (197.67,89.5) .. controls (199.61,89.5) and (201.17,91.07) .. (201.17,93) .. controls (201.17,94.93) and (199.61,96.5) .. (197.67,96.5) .. controls (195.74,96.5) and (194.17,94.93) .. (194.17,93) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (258.63,209) .. controls (258.63,207.07) and (260.2,205.5) .. (262.13,205.5) .. controls (264.07,205.5) and (265.63,207.07) .. (265.63,209) .. controls (265.63,210.93) and (264.07,212.5) .. (262.13,212.5) .. controls (260.2,212.5) and (258.63,210.93) .. (258.63,209) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (129.5,209) .. controls (129.5,207.07) and (131.07,205.5) .. (133,205.5) .. controls (134.93,205.5) and (136.5,207.07) .. (136.5,209) .. controls (136.5,210.93) and (134.93,212.5) .. (133,212.5) .. controls (131.07,212.5) and (129.5,210.93) .. (129.5,209) -- cycle ;
\draw  [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 }  ,fill opacity=1 ] (452.17,52) .. controls (452.17,50.07) and (453.74,48.5) .. (455.67,48.5) .. controls (457.61,48.5) and (459.17,50.07) .. (459.17,52) .. controls (459.17,53.93) and (457.61,55.5) .. (455.67,55.5) .. controls (453.74,55.5) and (452.17,53.93) .. (452.17,52) -- cycle ;
\draw  [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][fill={rgb, 255:red, 248; green, 231; blue, 28 }  ,fill opacity=1 ] (452.17,93) .. controls (452.17,91.07) and (453.74,89.5) .. (455.67,89.5) .. controls (457.61,89.5) and (459.17,91.07) .. (459.17,93) .. controls (459.17,94.93) and (457.61,96.5) .. (455.67,96.5) .. controls (453.74,96.5) and (452.17,94.93) .. (452.17,93) -- cycle ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (417,149.2) -- (486,218.2) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (421,145.2) -- (490,214.2) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (417,214.2) -- (486,145.2) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (421,218.2) -- (490,149.2) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (417,145.2) -- (490,145.2) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (417,218.2) -- (490,218.2) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (490,145.2) -- (490,218.2) ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (486,145.2) .. controls (486,142.99) and (487.79,141.2) .. (490,141.2) .. controls (492.21,141.2) and (494,142.99) .. (494,145.2) .. controls (494,147.41) and (492.21,149.2) .. (490,149.2) .. controls (487.79,149.2) and (486,147.41) .. (486,145.2) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (486,218.2) .. controls (486,215.99) and (487.79,214.2) .. (490,214.2) .. controls (492.21,214.2) and (494,215.99) .. (494,218.2) .. controls (494,220.41) and (492.21,222.2) .. (490,222.2) .. controls (487.79,222.2) and (486,220.41) .. (486,218.2) -- cycle ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][fill={rgb, 255:red, 248; green, 231; blue, 28 }  ,fill opacity=1 ][line width=1.5]    (417,145.2) -- (417,218.2) ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (413,218.2) .. controls (413,215.99) and (414.79,214.2) .. (417,214.2) .. controls (419.21,214.2) and (421,215.99) .. (421,218.2) .. controls (421,220.41) and (419.21,222.2) .. (417,222.2) .. controls (414.79,222.2) and (413,220.41) .. (413,218.2) -- cycle ;
\draw  [color={rgb, 255:red, 0; green, 0; blue, 0 }  ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (413,145.2) .. controls (413,142.99) and (414.79,141.2) .. (417,141.2) .. controls (419.21,141.2) and (421,142.99) .. (421,145.2) .. controls (421,147.41) and (419.21,149.2) .. (417,149.2) .. controls (414.79,149.2) and (413,147.41) .. (413,145.2) -- cycle ;
\draw   (321,142) -- (335.19,142) -- (335.19,139) -- (344.66,145) -- (335.19,151) -- (335.19,148) -- (321,148) -- cycle ;
\draw (395,139) node [anchor=north west][inner sep=0.75pt]   [align=left] {3};
\draw (500,139) node [anchor=north west][inner sep=0.75pt]   [align=left] {4};
\draw (395,205) node [anchor=north west][inner sep=0.75pt]   [align=left] {$3'$};
\draw (500,205) node [anchor=north west][inner sep=0.75pt]   [align=left] {$4'$};
\draw (205,85) node [anchor=north west][inner sep=0.75pt]   [align=left] {1};
\draw (115,200) node [anchor=north west][inner sep=0.75pt]   [align=left] {3};
\draw (205,155) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
\draw (266,200) node [anchor=north west][inner sep=0.75pt]   [align=left] {4};
\draw (465,45) node [anchor=north west][inner sep=0.75pt]   [align=left] {1};
\draw (465,85) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
\end{tikzpicture}
\caption{$K_4^3$ and $K_4^3\boxplus \{3, 4\}$.}
\end{figure}




\begin{definition}
Let $\mathcal{G}$ be a $3$-graph.
A pair $\{v_1,v_2\}\subset V(\mathcal{G})$ is symmetric in $\mathcal{G}$ if
$$L_{\mathcal{G}}(v_1) - v_2 = L_{\mathcal{G}}(v_2) - v_1.$$
\end{definition}

The crossed blowup of a $3$-graph has the following properties.

\begin{proposition}\label{PROP:nonminimal-corss-blowup}
Suppose that $\mathcal{G}$ is an $m$-vertex $3$-graph and $\{v_1,v_2\}\subset V(\mathcal{G})$ is a pair of vertices with $d(v_1,v_2)\ge 2$.
Then the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item The $3$-graph $\mathcal{G}$ is contained in $\mathcal{G}\boxplus\{v_1,v_2\}$ as an induced subgraph.
        In particular, $\lambda(\mathcal{G}) \le \lambda(\mathcal{G}\boxplus\{v_1,v_2\})$.
\item The $3$-graph $\mathcal{G}\boxplus\{v_1,v_2\}$ is $2$-covered iff $\mathcal{G}$ is $2$-covered.
\item If $\{v_1,v_2\}$ is symmetric in $\mathcal{G}$,
        then $\lambda(\mathcal{G}\boxplus\{v_1,v_2\}) = \lambda(\mathcal{G})$.
        If, in addition, there exists $(x_1,\ldots,x_{m})\in Z(\mathcal{G})$ with $x_1 + x_2 > 0$,
        then the set $Z(\mathcal{G}\boxplus\{v_1,v_2\})$ contains a one-dimensional simplex (i.e. a nontrivial line segment).
\end{enumerate}
\end{proposition}

\textbf{Remarks.}
\begin{itemize}
\item It is easy to see that Proposition~\ref{PROP:nonminimal-corss-blowup}~(b) does not necessarily
hold for the ordinary blowup of $\mathcal{G}$.

\item The requirement that $\{v_1,v_2\}$ is symmetric in
Proposition~\ref{PROP:nonminimal-corss-blowup}~(c)
cannot be removed since some calculations show that
$\lambda(C_{5}^{3}) = 1/25$ while $\lambda(C_{5}^{3}\boxplus\{4,5\}) = 4/81$,
where $C_{5}^{3} = \{123,234,345,451,512\}$.
\end{itemize}

\begin{figure}[htbp]
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (393.5,210) -- (449,168.5) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (397,213.5) -- (452.5,172) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (397,168.5) -- (452.5,210) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (393.5,172) -- (449,213.5) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (393.5,168.5) -- (452.5,168.5) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=1.5]    (452.52,213.26) -- (393.5,213.5) ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.2 ]   (478.5,122.5) -- (423.82,88.97) -- (452.5,213.5) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.2 ]   (368.5,122.5) -- (423.58,89.38) -- (393.5,213.5) -- cycle ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (393.5,213.5) -- (393.5,168.5) ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=1.5]    (452.52,213.26) -- (452.52,168.26) ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.4 ]   (478.5,122.5) -- (423.58,89.38) -- (452.5,168.5) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.4 ]   (368.5,122.5) -- (423.5,89.5) -- (393.5,168.5) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.3 ]   (195.07,107) -- (244.58,140.86) -- (225.67,195.64) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.3 ]   (145.55,140.86) -- (195.07,107) -- (244.58,140.86) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.3 ]   (145.55,140.86) -- (164.46,195.64) -- (195.07,107) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.3 ]   (164.46,195.64) -- (225.67,195.64) -- (145.55,140.86) -- cycle ;
\draw [draw opacity=0][fill=uuuuuu,fill opacity=0.3 ]   (225.67,195.64) -- (164.46,195.64) -- (244.58,140.86) ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (191.57,107) .. controls (191.57,105.07) and (193.13,103.5) .. (195.07,103.5) .. controls (197,103.5) and (198.57,105.07) .. (198.57,107) .. controls (198.57,108.93) and (197,110.5) .. (195.07,110.5) .. controls (193.13,110.5) and (191.57,108.93) .. (191.57,107) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (241.08,140.86) .. controls (241.08,138.93) and (242.65,137.36) .. (244.58,137.36) .. controls (246.52,137.36) and (248.08,138.93) .. (248.08,140.86) .. controls (248.08,142.79) and (246.52,144.36) .. (244.58,144.36) .. controls (242.65,144.36) and (241.08,142.79) .. (241.08,140.86) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (222.17,195.64) .. controls (222.17,193.71) and (223.74,192.14) .. (225.67,192.14) .. controls (227.6,192.14) and (229.17,193.71) .. (229.17,195.64) .. controls (229.17,197.57) and (227.6,199.14) .. (225.67,199.14) .. controls (223.74,199.14) and (222.17,197.57) .. (222.17,195.64) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (142.05,140.86) .. controls (142.05,138.93) and (143.62,137.36) .. (145.55,137.36) .. controls (147.48,137.36) and (149.05,138.93) .. (149.05,140.86) .. controls (149.05,142.79) and (147.48,144.36) .. (145.55,144.36) .. controls (143.62,144.36) and (142.05,142.79) .. (142.05,140.86) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (160.96,195.64) .. controls (160.96,193.71) and (162.53,192.14) .. (164.46,192.14) .. controls (166.4,192.14) and (167.96,193.71) .. (167.96,195.64) .. controls (167.96,197.57) and (166.4,199.14) .. (164.46,199.14) .. controls (162.53,199.14) and (160.96,197.57) .. (160.96,195.64) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (420,89.5) .. controls (420,87.57) and (421.57,86) .. (423.5,86) .. controls (425.43,86) and (427,87.57) .. (427,89.5) .. controls (427,91.43) and (425.43,93) .. (423.5,93) .. controls (421.57,93) and (420,91.43) .. (420,89.5) -- cycle ;
\draw  [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 }  ,fill opacity=1 ] (365,122.5) .. controls (365,120.57) and (366.57,119) .. (368.5,119) .. controls (370.43,119) and (372,120.57) .. (372,122.5) .. controls (372,124.43) and (370.43,126) .. (368.5,126) .. controls (366.57,126) and (365,124.43) .. (365,122.5) -- cycle ;
\draw  [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][fill={rgb, 255:red, 248; green, 231; blue, 28 }  ,fill opacity=1 ] (475,122.5) .. controls (475,120.57) and (476.57,119) .. (478.5,119) .. controls (480.43,119) and (482,120.57) .. (482,122.5) .. controls (482,124.43) and (480.43,126) .. (478.5,126) .. controls (476.57,126) and (475,124.43) .. (475,122.5) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (390,168.5) .. controls (390,166.57) and (391.57,165) .. (393.5,165) .. controls (395.43,165) and (397,166.57) .. (397,168.5) .. controls (397,170.43) and (395.43,172) .. (393.5,172) .. controls (391.57,172) and (390,170.43) .. (390,168.5) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (449,168.5) .. controls (449,166.57) and (450.57,165) .. (452.5,165) .. controls (454.43,165) and (456,166.57) .. (456,168.5) .. controls (456,170.43) and (454.43,172) .. (452.5,172) .. controls (450.57,172) and (449,170.43) .. (449,168.5) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (390,213.5) .. controls (390,211.57) and (391.57,210) .. (393.5,210) .. controls (395.43,210) and (397,211.57) .. (397,213.5) .. controls (397,215.43) and (395.43,217) .. (393.5,217) .. controls (391.57,217) and (390,215.43) .. (390,213.5) -- cycle ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (449,213.5) .. controls (449,211.57) and (450.57,210) .. (452.5,210) .. controls (454.43,210) and (456,211.57) .. (456,213.5) .. controls (456,215.43) and (454.43,217) .. (452.5,217) .. controls (450.57,217) and (449,215.43) .. (449,213.5) -- cycle ;
\draw   (298,158) -- (312.19,158) -- (312.19,155) -- (321.66,161) -- (312.19,167) -- (312.19,164) -- (298,164) -- cycle ;

\draw (430,75) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
\draw (485,115) node [anchor=north west][inner sep=0.75pt]   [align=left] {3};
\draw (350,115) node [anchor=north west][inner sep=0.75pt]   [align=left] {1};
\draw (375,162) node [anchor=north west][inner sep=0.75pt]   [align=left] {5};
\draw (460,162) node [anchor=north west][inner sep=0.75pt]   [align=left] {4};
\draw (375,205) node [anchor=north west][inner sep=0.75pt]   [align=left] {$5'$};
\draw (460,205) node [anchor=north west][inner sep=0.75pt]   [align=left] {$4'$};
\draw (190,86) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
\draw (250,132) node [anchor=north west][inner sep=0.75pt]   [align=left] {3};
\draw (130,132) node [anchor=north west][inner sep=0.75pt]   [align=left] {1};
\draw (231,198) node [anchor=north west][inner sep=0.75pt]   [align=left] {4};
\draw (150,198) node [anchor=north west][inner sep=0.75pt]   [align=left] {5};
\end{tikzpicture}
\caption{$C_{5}^3$ and $C_{5}^3\boxplus \{4,5\}$.}
\end{figure}

Using Propositions~\ref{PROP:nonminal-finite-family} and~\ref{PROP:nonminimal-corss-blowup} and some further argument
we obtain the following theorem.

\begin{theorem}\label{THM:nonminimal-general-turan-stability}
Let $\mathcal{G}$ be a minimal $3$-graph.
Suppose that $\{v_1,v_2\}\subset V(\mathcal{G})$ is symmetric in $\mathcal{G}$ and $d(v_1,v_2)\ge 1$.
Then there exists a finite family $\mathcal{F}$ of $3$-graphs with $\xi(\mathcal{F}) = \infty$
such that $\pi(\mathcal{F}) = 6 \lambda(\mathcal{G})$,
and for every $n\in\mathbb{N}$ there exists a maximum $\mathcal{F}$-free $3$-graph
on $n$-vertices which is a blowup of $\mathcal{G}$.
\end{theorem}
\subsection{Feasible region}\label{SUBSEC:feasible-region}
Theorem~\ref{THM:main-sec1.1} has some interesting application to the so-called feasible region of hypergraphs.

Given an $r$-graph $\mathcal{H}$ the \emph{shadow} of $\mathcal{H}$ is defined as
\begin{align}
\partial\mathcal{H}
=
\left\{A\in \binom{V(\mathcal{H})}{r-1}\colon \text{there is } B\in \mathcal{H} \text{ such that }
	A\subseteq B\right\}. \notag
\end{align}
The \emph{edge density} of $\mathcal{H}$ is defined as $\rho(\mathcal{H})= |\mathcal{H}|/\binom{v(\mathcal{H})}{r}$, and
the \emph{shadow density} of $\mathcal{H}$ is defined as $\rho(\partial\mathcal{H})= |\partial\mathcal{H}|/\binom{v(\mathcal{H})}{r-1}$.

For a family $\mathcal{F}$ the \emph{feasible region} $\Omega(\mathcal{F})$ of $\mathcal{F}$
is the set of points $(x,y)\in [0,1]^2$ such that there exists a sequence of $\mathcal{F}$-free $r$-graphs
$\left( \mathcal{H}_{k}\right)_{k=1}^{\infty}$ with
$$\lim_{k \to \infty}v(\mathcal{H}_{k}) = \infty,\quad
\lim_{k \to \infty}\rho(\partial\mathcal{H}_{k}) = x, \quad\text{and}\quad \lim_{k \to \infty}\rho(\mathcal{H}_{k}) = y.$$
The feasible region unifies and generalizes many classical problems
such as the  Kruskal--Katona theorem~\cite{KR63,KA68} and the Tur\'{a}n problem.
It was introduced in \cite{LM1} to understand the extremal properties of $\mathcal{F}$-free hypergraphs
beyond just the determination of $\pi(\mathcal{F})$. The general shape of $\Omega(\mathcal{F})$ was analyzed
in~\cite{LM1} as follows: For some constant $c(\mathcal{F})\in [0, 1]$ the projection to the first
coordinate,
\[
	{\rm proj}\Omega(\mathcal{F})
	=
	\left\{ x \colon  \text{there is $y \in [0,1]$ such that $(x,y) \in \Omega(\mathcal{F})$} \right\},
\]
is the interval $[0, c(\mathcal{F})]$ .
Moreover, there is a left-continuous almost everywhere differentiable function
$g(\mathcal{F})\colon {\rm proj}\Omega(\mathcal{F}) \to [0,1]$ such that
\[
	\Omega(\mathcal{F})
	=
	\bigl\{(x, y)\in [0, c(\mathcal{F})]\times [0, 1]\colon 0\le y\le g(\mathcal{F})(x)\bigr\}\,.
\]
Let us call $g(\mathcal{F})$ the \emph{feasible region function} of $\mathcal{F}$.
It was shown in~\cite{LM1} that $g(\mathcal{F})$ is not necessarily continuous,
and in~\cite{Liu20a}, it was shown that $g(\mathcal{F})$ can
have infinitely many local maxima even for some simple and natural families $\mathcal{F}$.

The stability number of $\mathcal{F}$ can give information about the number of global maxima of $g(\mathcal{F})$ (e.g. see~\cite{LMR1}).
The family $\mathcal{M}$ of triple systems from~\cite{LM22} for which $\xi(\mathcal{M})=2$ has the following
additional property: not only are the two near extremal constructions for $\mathcal{M}$ far from each
other in edit-distance, but the same is true of their shadows.
As a consequence, in addition to $\xi(\mathcal{M}) = 2$, the function~$g(\mathcal{M})$ has exactly two global maxima.
Similarly, it was proved in~\cite{LMR1} that for the $t$-stable family $\mathcal{M}_t$
the function $g(\mathcal{M}_t)$ has exactly $t$ global maxima for every positive integer $t$.

It was left as an open question in~\cite{LMR1}
whether there exists a finite family $\mathcal{F}$
so that the function $g(\mathcal{F})$ has infinitely many global maxima.
Here we would like to remind the reader that even though there are
infinitely many extremal constructions for Tur\'{a}n's tetrahedron conjecture (if it is true),
the shadow of all these constructions is complete.
Hence, solving Tur\'{a}n's tetrahedron conjecture will not answer the question asked in~\cite{LMR1}.

Our next result shows that the same family $\mathcal{F}_t$ as in Theorem~\ref{THM:main-sec1.1}
has the property that $g(\mathcal{F}_t)$ attains its maximum on a nontrivial interval,
thus giving a positive answer to the question in~\cite{LMR1}.

\begin{theorem}\label{THM:feasibe-region}
For every integer $t\ge 3$ we have $\mathrm{proj}\Omega(\mathcal{F}_t) = \left[0, \frac{t+1}{t+2}\right]$,
and $g(\mathcal{F}_t, x)\leq {(t-2)(t-1)}/{t^2}$ for all $x\in \mathrm{proj}\Omega(\mathcal{F}_t)$.
Moreover, if $t\ge 4$, then $g(\mathcal{F}_t, x)={(t-2)(t-1)}/{t^2}$
iff $x\in \left[\frac{t-1}{t}, \frac{t-1}{t}+\frac{1}{t^2}\right]$.
\end{theorem}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[xscale=8,yscale=5]
\draw [->] (0,0)--(1+0.05,0);
\draw [->] (0,0)--(0,1+0.05);
\draw (0,1)--(1,1);
\draw (1,0)--(1,1);
\draw [line width=0.8pt,dash pattern=on 1pt off 1.2pt,domain=0:1] plot(\x,1-0.05);
\draw [line width=0.4pt,dash pattern=on 1pt off 1.2pt] (1/2,0) -- (1/2,1-0.05);
\draw [line width=0.4pt,dash pattern=on 1pt off 1.2pt] (2/3,0) -- (2/3,1-0.05);
\draw [line width=0.4pt,dash pattern=on 1pt off 1.2pt] (5/6,0) -- (5/6,1-0.05);


\draw[fill=sqsqsq,fill opacity=0.25, line width=0.5pt]
(0,0)  to [out = 10, in = 260] (1/2,1-0.05) to (2/3, 1-0.05)
to [out = 280, in = 140] (5/6,4/9) to (5/6,0);

\begin{scriptsize}
\draw [fill=uuuuuu] (1,0) circle (0.1pt);
\draw[color=uuuuuu] (1,0-0.05) node {$1$};
\draw [fill=uuuuuu] (0,0) circle (0.1pt);
\draw[color=uuuuuu] (0-0.03,0-0.03) node {$0$};
\draw [fill=uuuuuu] (0,1) circle (0.1pt);
\draw[color=uuuuuu] (0-0.05,1+0.05) node {$y$};
\draw[color=uuuuuu] (1+0.05,0-0.05) node {$x$};
\draw [fill=uuuuuu] (0,1-0.05) circle (0.1pt);
\draw[color=uuuuuu] (0-0.1,1-0.05) node {$\frac{(t-2)(t-1)}{t^2}$};

\draw [fill=uuuuuu] (1/2,1-0.05) circle (0.1pt);
\draw [fill=uuuuuu] (1/2,0) circle (0.1pt);
\draw[color=uuuuuu] (1/2,0-0.06) node {$\frac{t-1}{t}$};

\draw [fill=uuuuuu] (2/3,1-0.05) circle (0.1pt);
\draw [fill=uuuuuu] (2/3,0) circle (0.1pt);
\draw[color=uuuuuu] (2/3,0-0.06) node {$\frac{t-1}{t}+\frac{1}{t^2}$};

\draw [fill=uuuuuu] (5/6,1-0.05) circle (0.1pt);
\draw [fill=uuuuuu] (5/6,0) circle (0.1pt);
\draw[color=uuuuuu] (5/6,0-0.06) node {$\frac{t+1}{t+2}$};

\draw [fill=uuuuuu] (1,1) circle (0.1pt);
\draw[color=uuuuuu] (1+0.04,1+0.04) node {$(1,1)$};

\draw [fill=uuuuuu] (5/6,4/9) circle (0.1pt);
\end{scriptsize}
\end{tikzpicture}
\caption{The function $g(\mathcal{F}_t)$ attains its maximum on the interval $\left[\frac{t-1}{t}, \frac{t-1}{t}+\frac{1}{t^2}\right]$.}
\end{figure}


The remainder of this paper is organized as follows.
In Section~\ref{SEC:Preliminaries} we introduce some preliminary definitions and results.
In Section~\ref{SEC:Proof-multilinear-polynomials} we prove Proposition~\ref{PROP:multilinear-polynomial}.
In Section~\ref{SEC:Proof-nonminimal-hypergraphs} we prove Propositions~\ref{PROP:nonminal-finite-family} and~\ref{PROP:nonminimal-corss-blowup} and Theorem~\ref{THM:nonminimal-general-turan-stability}.
In Section~\ref{SEC:proof-turan-number} we prove Theorem~\ref{THM:main-sec1.1}~(a),~(b), and~(c).
In Section~\ref{SEC:proof-stability} we prove Theorem~\ref{THM:main-sec1.1}~(d).
In Section~\ref{SEC:proof-feasible-region} we prove Theorem~\ref{THM:feasibe-region}.
The final section conatins some concluding remarks.

\section{Preliminaries}\label{SEC:Preliminaries}
We present some definitions and useful results in this section.

Recall that the shadow of an $r$-graph $\mathcal{H}$ is
\begin{align}
\partial\mathcal{H}
=
\left\{A\in \binom{V(\mathcal{H})}{r-1}\colon \text{there is } B\in \mathcal{H} \text{ such that }
	A\subseteq B\right\}. \notag
\end{align}
By setting $\partial_1\mathcal{H} = \partial\mathcal{H}$
for every $i \in [r-1]$ we define $\partial_{i}\mathcal{H}$ inductively
as $\partial_{i}\mathcal{H} = \partial \partial_{i-1}\mathcal{H}$.
In particular, $\partial_{r-2}\mathcal{H}$ is a graph.

Given an $r$-graph $\mathcal{H}$ and a vertex $v\in V(\mathcal{H})$
the \emph{neighborhood} of $v$ is defined as
\[
N_{\mathcal{H}}(v)
= \left\{u\in V(\mathcal{H})\setminus\{v\}\colon
     \exists E\in\mathcal{H}\text{ such that }\{u,v\}\subset E\right\}.
\]
Recall that the \emph{link} of $v$ in $\mathcal{H}$ is
$L_\mathcal{H}(v)
=\left\{e\in \partial\mathcal{H}\colon
     e\cup \{v\}\in \mathcal{H}\right\}.$
The \emph{degree} of $v$ is $d_{\mathcal{H}}(v)=|L_\mathcal{H}(v)|$.
Denote by $\delta(\mathcal{H})$ and $\Delta(\mathcal{H})$
the minimum degree and maximum degree of $\mathcal{H}$, respectively.
Recall that the neighborhood of $\{u,v\}\subset V(\mathcal{H})$ in $\mathcal{H}$ is
$N_{\mathcal{H}}(u,v)
=\left\{w\in V(\mathcal{H})\colon \{u,v,w\}\in \mathcal{H}\right\}.$
The codegree of  $\{u,v\}$ is $d_{\mathcal{H}}(u,v) = |N_{\mathcal{H}}(u,v)|$.
Denote by $\delta_2(\mathcal{H})$ and $\Delta_2(\mathcal{H})$
the minimum degree and maximum codegree of $\mathcal{H}$, respectively.
We will omit the subscript $\mathcal{H}$ from our notations if it is clear from the context.

Let $\mathcal{G}$ be an $r$-graph with the vertex set $[m]$, and $V_1,\ldots, V_m$ be $m$ disjoint sets
(each $V_i$ is allowed to be empty).
The blowup $\mathcal{G}[V_1,\ldots, V_m]$ of $\mathcal{G}$ is obtained from $\mathcal{G}$
by replacing each vertex $i$ with the set $V_i$ and replacing each edge $i_1\cdots i_r$
with the complete $r$-partite $r$-graph with parts $V_{i_1},\ldots,V_{i_r}$.

For a hypergraph $\mathcal{G}$ the maximum number of edges in a blowup of $\mathcal{G}$
is related to $\lambda(\mathcal{G})$
(e.g. see Frankl and F\"uredi~\cite{FF89} or Keevash's survey~\cite[{Section~3}]{KE11}).

\begin{lemma}[\cite{FR84,KE11}]\label{LEMMA:|H|<=lambda-T-n^r}
Suppose that $\mathcal{G}$ is an $r$-graph and $V_1,\ldots, V_m$ are $m$ disjoint sets with $\sum_{i\in[m]}|V_i| = n$.
Then $|\mathcal{G}[V_1,\ldots, V_m]| = p_{\mathcal{G}}\left(|V_1|/n,\ldots,|V_m|/n\right)n^r$.
In particular, $|\mathcal{G}[V_1,\ldots, V_m]|\le \lambda(G)n^r$.
\end{lemma}


Given an $r$-graph $F$ we say $\mathcal{H}$ is \emph{$F$-hom-free}
if there is no homomorphism from $F$ to $\mathcal{H}$.
This is equivalent to say that every blowup of $\mathcal{H}$ is $F$-free.
For a family $\mathcal{F}$ of $r$-graphs we say $\mathcal{H}$ is \emph{$\mathcal{F}$-hom-free}
if it is $F$-hom-free for all $F \in \mathcal{F}$.
An easy observation is that
if an $r$-graph $F$ is $2$-covered, then $\mathcal{H}$ is $F$-free iff it is $F$-hom-free.

\begin{definition}[Blowup-invariance]
A family $\mathcal{F}$ of $r$-graphs is blowup-invariant if every
$\mathcal{F}$-free $r$-graph is also $\mathcal{F}$-hom-free.
\end{definition}

Let $\mathcal{H}$ be an $r$-graph and $\{u,v\}\subset V(\mathcal{H})$ be two non-adjacent vertices
(i.e., no edge contains both $u$ and $v$).
We say $u$ and $v$ are \emph{equivalent} if $L_{\mathcal{H}}(u)=L_{\mathcal{H}}(v)$.
Otherwise we say they are \emph{non-equivalent}.
An equivalence class of $\mathcal{H}$ is a maximal vertex set in which every pair of vertices are equivalent.
We say $\mathcal{H}$ is \emph{symmetrized} if it does not contain non-equivalent pairs of vertices.
In~\cite{LMR2}, the authors summarized the well known method of Zykov~\cite{Zy} symmetrization for solving Tur\'{a}n problems
into the following statement.

\begin{theorem}[see e.g.~\cite{LMR2}]\label{THEOREM:ex(n,F)=max-blowup-of-H}
Suppose that $\mathcal{F}$ is a blowup-invariant family of $r$-graphs. If $\mathfrak{H}$ denotes the class of all symmetrized $\mathcal{F}$-free $r$-graphs, then ex$(n, \mathcal{F})=\mathfrak{h}(n)$  holds for every $n \in \mathbb{N}^{+}$,
where $\mathfrak{h}(n)=\max\{|\mathcal{H}|\colon\mathcal{H}\in \mathfrak{H} \text{ and } v(\mathcal{H})=n\}$.
\end{theorem}
\section{Multilinear polynomials}\label{SEC:Proof-multilinear-polynomials}
We prove Proposition~\ref{PROP:multilinear-polynomial} in this section.
First, we need the following simple lemma.

\begin{lemma}\label{LEMMA:max-p-symmetric}
Suppose that $p(X_1,\ldots,X_m) = p_1 + p_2(X_i+X_j) + p_3X_iX_j$
is an $m$-variable multilinear polynomial that is symmetric with respect to $X_i$ and $X_j$,
and $p_3$ is nonnegative.
Then for every $(x_1, \ldots, x_m)\in Z(p)$ we have
$(x_1, \ldots,x_{i-1}, (x_i+x_j)/2, x_{i+1}, \ldots, x_{j-1}, (x_i+x_j)/2, x_{j+1}, \ldots, x_m)\in Z(p)$.
In particular,
$$\max\{p(y_1, \ldots, y_m)\colon (y_1, \ldots, y_m)\in \Delta_{m-1} \text{ and }y_i = y_j\} = \lambda(p).$$
\end{lemma}
\begin{proof}
Using the AM-GM inequality we obtain
\begin{align}
& p(x_1, \ldots,x_{i-1}, \frac{x_i+x_j}{2}, x_{i+1}, \ldots, x_{j-1}, \frac{x_i+x_j}{2}, x_{j+1}, \ldots, x_m) \notag\\
& = p_1 + p_2\left(\frac{x_i+x_j}{2}+ \frac{x_i+x_j}{2}\right) + p_3\left(\frac{x_i+x_j}{2}\right)^2  \notag\\
& \ge p_1 + p_2(x_i+x_j) + p_3 x_ix_j
= \lambda(p), \notag
\end{align}
where the last equality follows from our assumption that $(x_1, \ldots, x_m)\in Z(p)$.
By the definition of $\lambda(p)$, the inequality above is actually an equality, so we have
$(x_1, \ldots,x_{i-1}, (x_i+x_j)/2, x_{i+1}, \ldots, x_{j-1}, (x_i+x_j)/2, x_{j+1}, \ldots, x_m)\in Z(p)$.
\end{proof}

Now we are ready to prove Proposition~\ref{PROP:multilinear-polynomial}.

\begin{proof}[Proof of Proposition~\ref{PROP:multilinear-polynomial}]
Let $p, p_1, p_2, p_3, p_4, p_5, \hat{p}$ be polynomials that satisfy the assumptions in Proposition~\ref{PROP:multilinear-polynomial}.
By symmetry, we may assume that $\{i,j\} = \{1,2\}$.
Let $\widehat{X}_1 = X_1+ X_1'$ and $\widehat{X}_2 = X_2+X_2'$.
It follows from the AM-GM inequality that
\begin{align}\label{equ:hat-p-and-p}
& \hat{p}(X_1,X'_1,X_2,X'_2,X_3,\ldots,X_m)  \notag\\
&  = p_1 + p_2(X_1+X'_1+X_2+X'_2)
            + p_{4}(X_1+X'_1)(X_2+X'_2)
            + p_{5}(X_1+X_2)(X'_1+X'_2) \notag\\
& \le p_1 + p_2(\widehat{X}_1+\widehat{X}_2)
            + p_{4}\left(\frac{\widehat{X}_1+\widehat{X}_2}{2}\right)^2
            + p_{5}\left(\frac{\widehat{X}_1+\widehat{X}_2}{2}\right)^2 \notag\\
& = p_1 + p_2(\widehat{X}_1+\widehat{X}_2)
            + p_{3}\left(\frac{\widehat{X}_1+\widehat{X}_2}{2}\right)^2 \notag\\
& = p((\widehat{X}_1+\widehat{X}_2)/2, (\widehat{X}_1+\widehat{X}_2)/2, X_3, \ldots, X_m).
\end{align}
This implies that $\lambda(\hat{p}) \le \lambda(p)$.

Let $\alpha \in [0,1]$ and
\begin{align}
\vec{y} = \alpha \cdot (\frac{x_1+x_2}{2},0,0,\frac{x_1+x_2}{2}, x_3, \ldots, x_m)
+ (1-\alpha) \cdot (0,\frac{x_1+x_2}{2},\frac{x_1+x_2}{2},0, x_3, \ldots, x_m). \notag
\end{align}
Then it follows from Equation~(\ref{equ:hat-p-and-p}) that
\begin{align}
 \hat{p}(\vec{y})  = p((x_1+x_2)/2, (x_1+x_2)/2, x_3, \ldots, x_m). \notag
\end{align}
By Lemma~\ref{LEMMA:max-p-symmetric}, we have
\begin{align}
p((x_1+x_2)/2, (x_1+x_2)/2, x_3, \ldots, x_m)
= p(x_1, x_2, x_3, \ldots, x_m) = \lambda(p). \notag
\end{align}
Therefore, $\vec{y}\in Z(\hat{p})$.
\end{proof}
\section{Nonmininal hypergraphs}\label{SEC:Proof-nonminimal-hypergraphs}
In this section, we prove results that were stated in Section~\ref{SUBSEC:nonminimal-hypergraphs}.

\subsection{Proof for Proposition~\ref{PROP:nonminal-finite-family}}
In this subsection we prove Proposition~\ref{PROP:nonminal-finite-family}.

For every $r$-graph $\mathcal{G}$ let $\mathcal{F}_{\infty}(\mathcal{G})$ be the (infinite) family of all $r$-graphs that are not $\mathcal{G}$-colorable, i.e.
\begin{align}
\mathcal{F}_{\infty}(\mathcal{G})
= \left\{\text{$r$-graph $F$} \colon \text{ and $F$ is not $\mathcal{G}$-colorable}\right\}.\notag
\end{align}
For every positive integer $M$ define the family $\mathcal{F}_{M}(\mathcal{G})$
of $r$-graphs as
\begin{align}
\mathcal{F}_{M}(\mathcal{G})
= \left\{F \in \mathcal{F}_{\infty}(\mathcal{G})\colon v(F)\le M \right\}.\notag
\end{align}
It is clear that $\mathcal{F}_{M}(\mathcal{G})$ is nonempty if $M\ge v(\mathcal{G})+1$
since it contains all $r$-graphs $F$ with at most $M$ vertices and $K_{v(\mathcal{G})+1}\subset \partial_{r-2}F$.
It is also clear from the definition that $\mathcal{F}_{M}(\mathcal{G}) \subset \mathcal{F}_{M'}(\mathcal{G})$ for all $M'\ge M$.

It follows easily from the definition that every $\mathcal{F}_{\infty}(\mathcal{G})$-free $r$-graph is $\mathcal{G}$-colorable.
In particular, every maximum $\mathcal{F}_{\infty}(\mathcal{G})$-free $r$-graph on $n$ vertices is $\mathcal{G}$-colorable.
One of the main result in~\cite{PI14} implies the following compactness result.
For every minimum $r$-graph $\mathcal{G}$ there exists a constant $M_0 \in \mathbb{N}$ such that every maximum $\mathcal{F}_{\infty}(\mathcal{G})$-free $r$-graph on $n$ vertices is $\mathcal{G}$-colorable.
Proposition~\ref{PROP:nonminal-finite-family} actually shows that a similar compactness result holds for all $r$-graphs $\mathcal{G}$.

The following simple lemma is a special case of Lemma~8 in~\cite{PI14}.

\begin{lemma}[\cite{PI14}]\label{LEMMA:F-free-equ-F-homo-free}
For every $r$-graph $\mathcal{G}$ and every positive integer $M$ the family $\mathcal{F}_{M}(\mathcal{G})$ is blowup-invariant.
\end{lemma}

The following lemma extends Observation~2.3~(a) in~\cite{LIU19}.

\begin{lemma}\label{LEMMA:F-free-shadow-Km-free}
Let $\mathcal{G}$ be an $r$-graph.
Let $m$ denote the order of a maximum complete subgraph in $\partial_{r-2}\mathcal{G}$
and let $M = m+1+(r-2)\binom{m+1}{2}$.
Suppose that an $r$-graph $\mathcal{H}$ is $\mathcal{F}_{M}(\mathcal{G})$-free.
Then $\partial_{r-2}\mathcal{H}$ is $K_{m+1}$-free.
\end{lemma}
\begin{proof}
Suppose to the contrary that there exists a set $C \subset V(\mathcal{H})$ of size $m+1$
such that the induced subgraph of $\partial\mathcal{H}$ on $C$ is complete.
For every pair $\{u,v\}\subset C$ let $E_{u,v}\in \mathcal{H}$ be an edge that contains $\{u,v\}$.
Let $F = \{E_{u,v}\colon \{u,v\}\subset C\}$.
It is clear that $v(F) \le m+1+(r-2)\binom{m+1}{2} = M$.
So it follows from our assumption that $F$ is $\mathcal{G}$-colorable.
In other words, there exists a homomorphism $\phi\colon V(F)\to V(\mathcal{G})$ from $F$ to $\mathcal{G}$.
Since every pair $\{u,v\}\subset C$ is contained in an edge of $F$, we have $\phi(u) \neq \phi(v)$.
Therefore, $|\phi(C)| = |C| = m+1$.
However, $u\sim v$ in $\partial_{r-2}F$ implies that $\phi(u)\sim \phi(v)$ in $\partial_{r-2}\mathcal{G}$,
which means that the induced subgraph of $\partial_{r-2}\mathcal{G}$ on $\phi(C)$ is complete, a contradiction.
\end{proof}

We will prove the following statement which implies
Proposition~\ref{PROP:nonminal-finite-family}.


\begin{proposition}\label{PROP:nonminal-finite-family+}
Suppose that $r\ge 3$, $\mathcal{G}$ is an $r$-graph, and $M = v(\mathcal{G})+1+(r-2)\binom{v(\mathcal{G})+1}{2}$.
Then $\mathrm{ex}(n,\mathcal{F}_{M}(\mathcal{G})) = \max\{|\mathcal{H}|\colon
\text{$v(\mathcal{H}) = n$ and $\mathcal{H}$ is $\mathcal{G}$-colorable}\}$
for every $n\in \mathbb{N}$.
\end{proposition}
\begin{proof}
Let $\mathcal{G}$ be an $r$-graph and let $m = v(\mathcal{G})$.
Let $\mathfrak{G}$ be the collection of all $\mathcal{G}$-colorable $r$-graphs.
Define $\mathfrak{g}(n) = \max\{|\mathcal{H}|\colon \mathcal{H} \in \mathfrak{H} \text{ and }v(\mathcal{H}) = n\}$.
Let $M = m+1 + (r-2)\binom{m+1}{2}$ and $\mathcal{F} = \mathcal{F}_{M}(\mathcal{G})$.
It follows from Lemma~\ref{LEMMA:F-free-equ-F-homo-free} that $\mathcal{F}$ is blowup-invariant.
Therefore, by Theorem~\ref{THEOREM:ex(n,F)=max-blowup-of-H},
it suffices to prove the following claim.

\begin{claim}\label{CLAIM:symmetrized-stable}
Every symmetrized $\mathcal{F}$-free $r$-graph is contained in $\mathfrak{G}$.
\end{claim}
\begin{proof}
Let $\mathcal{H}$ be a symmetrized $\mathcal{F}$-free $r$-graph.
Let $C\subset V(\mathcal{H})$ be a set that contains exactly one vertex from each equivalence class of $\mathcal{H}$
and let $\mathcal{T}$ denote the induced subgraph of $\mathcal{H}$ on $C$.
It is clear from the definition of symmetrized hypergraphs that $\mathcal{H}$ is a blowup of $\mathcal{T}$.
Since $\partial_{r-2}\mathcal{T} \cong K_{|C|}$, it follows from
Lemma~\ref{LEMMA:F-free-shadow-Km-free} that $|C|\le m$.
Since $\mathcal{T}\not\in \mathcal{F}$, we know that $\mathcal{T}$ is $\mathcal{G}$-colorable.
This implies that $\mathcal{H}$ is also $\mathcal{G}$-colorable.
Therefore, $\mathcal{H} \in \mathfrak{G}$.
\end{proof
This completes the proof of Proposition~\ref{PROP:nonminal-finite-family+}.
\end{proof

\subsection{Proofs for Proposition~\ref{PROP:nonminimal-corss-blowup} and Theorem~\ref{THM:nonminimal-general-turan-stability}}
In this subsection we prove Proposition~\ref{PROP:nonminimal-corss-blowup} and Theorem~\ref{THM:nonminimal-general-turan-stability}.

Let us assume that the vertex set of $\mathcal{G}$ is $\{v_1,v_2,\ldots, v_m\}$.
Let $v_1',v_2'$ denote the clones of $v_1,v_2$ in $\mathcal{G}\boxplus\{v_1,v_2\}$, respectively.
Let $U_1 = \{v_1,v_2',v_3,\ldots, v_m\}$ and $U_2 = \{v_1',v_2,v_3,\ldots, v_m\}$.
It is easy to see from the definition that the induced subgraph of $\mathcal{G}\boxplus\{v_1,v_2\}$ on
$U_1$ and $U_2$ are both isomorphic to $\mathcal{G}$, this proves~(a).
Statement~(b) follows easily from the definition of crossed blowup.
So we may focus on~(c).

We will prove the following statement which implies Proposition~\ref{PROP:nonminimal-corss-blowup}~(c).

\begin{proposition}\label{PROP:nonminimal-corss-blowup+}
Suppose that $\mathcal{G}$ is an $m$-vertex $3$-graph and $\{v_1,v_2\}\subset \mathcal{G}$ is a pair of vertices with $d(v_1,v_2)\ge 2$.
Suppose that $\{v_1,v_2\}$ is symmetric in $\mathcal{G}$.
Then $\lambda(\mathcal{G}\boxplus\{v_1,v_2\}) = \lambda(\mathcal{G})$.
Moreover, for every $(z_1,\ldots,z_{m})\in Z(\mathcal{G})$
we have
$$L\left((\bar{z},0,0,\bar{z},z_3,\ldots,z_{m}), (0,\bar{z},\bar{z},0,z_3,\ldots,z_{m})\right)
\subset Z(\mathcal{G}\boxplus\{v_1,v_2\}),$$
where $\bar{z} = (z_1+z_2)/2$.
\end{proposition}
\begin{proof}
Let $k = d(v_1,v_2)$ and assume that $N(v_1,v_2)=\{v_{i_1}, \ldots, v_{i_k}\}$.
Let $W = \{v_3,\ldots,v_m\}$. Define polynomials $p_1, \ldots, p_5$ as
\begin{align}
p_{1} & = \sum_{v_iv_jv_k\in \mathcal{G}[W]}X_iX_jX_k, \notag\\
p_{2} & = \sum_{v_iv_j\in L_{\mathcal{G}}(v_1)[W]}X_iX_j = \sum_{v_iv_j\in L_{\mathcal{G}}(v_2)[W]}X_iX_j,  \notag\\
p_{3} & = X_{i_1}+ \cdots + X_{i_k}, \quad
p_{4}  = X_{i_1}+ \cdots + X_{i_{k-1}}, \quad\text{and}\quad
p_{5} = X_{k}. \notag
\end{align}
It is easy to see that $p_{\mathcal{G}}(X_1, \ldots, X_{m})
 = p_1 + p_2(X_1 + X_2)+ p_{3}X_1X_2$.
On the other hand, it follows from the definition of crossed blowup that
\begin{align}
& p_{\mathcal{G}\boxplus\{v_1,v_2\}}(X_1,X_1',X_2,X_2',X_3 \ldots, X_{m}) \notag\\
& = p_1 + p_2(X_1 + X_2+X_1'+X_2')+ p_{4}(X_1+X_1')(X_2+X_2')+p_5(X_1+X_2)(X_1'+X_2').  \notag
\end{align}
So by Proposition~\ref{PROP:multilinear-polynomial} we have
$$L\left((\bar{z},0,0,\bar{z},z_3,\ldots,z_{m}), (0,\bar{z},\bar{z},0,z_3,\ldots,z_{m})\right)
\subset Z(\mathcal{G}\boxplus\{v_1,v_2\}).$$
\end{proof}

Now we are ready to prove Theorem~\ref{THM:nonminimal-general-turan-stability}.

\begin{proof}[Proof of Theorem~\ref{THM:nonminimal-general-turan-stability}]
If $d_{\mathcal{G}}(v_1,v_2) \ge 2$, then we let $\mathcal{G}' = \mathcal{G}\boxplus \{v_1,v_2\}$, and
it follows from Proposition~\ref{PROP:nonminimal-corss-blowup}~(c) that $\lambda(\mathcal{G}') = \lambda(\mathcal{G})$.

If $d_{\mathcal{G}}(v_1,v_2) = 1$, then we let $\{w\}= N_{\mathcal{G}}(v_1, v_2)$
and let $\mathcal{G}'$ be obtained as follows.
First, we take two vertex-disjoint copies of $\mathcal{G}$ and identify them on the set $V(\mathcal{G})\setminus \{w\}$
(i.e. we blowup $w$ into two vertices).
Denote by $\mathcal{G}^2$ the resulting $3$-graph.
For example, if $\mathcal{G} = \{125,345\}$ and $\{v_1,v_2\} = \{1,2\}$, then $\mathcal{G}^2 = \{125,345,126,346\}$.
Let $\mathcal{G}' = \mathcal{G}^2 \boxplus\{v_1,v_2\}$.
Since $\mathcal{G}^2$ is $\mathcal{G}$-colorable and $\mathcal{G}\subset \mathcal{G}^2$,
we have $\lambda(\mathcal{G}^2) = \lambda(\mathcal{G})$.
Notice that $\{v_1,v_2\}$ is still symmetric in $\mathcal{G}^2$,
so it follows from Proposition~\ref{PROP:nonminimal-corss-blowup}~(c) that
$\lambda(\mathcal{G}') = \lambda(\mathcal{G}^2) = \lambda(\mathcal{G})$.

Let $m = v(\mathcal{G}')$ and $M = m+1 + (r-2)\binom{m+1}{2}$.
Let $\mathcal{F} = \mathcal{F}_{M}(\mathcal{G}')$.
It follows from Proposition~\ref{PROP:nonminal-finite-family+} that
$\mathrm{ex}(n,\mathcal{F}) = \max\{|\mathcal{H}|\colon  \text{$v(\mathcal{H})=n$ and $\mathcal{H}$ is $\mathcal{G}'$-colorable}\}$.
Then by Lemma~\ref{LEMMA:|H|<=lambda-T-n^r}, we have  $\pi(\mathcal{F}) \le 6\lambda(\mathcal{G}') = 6\lambda(\mathcal{G})$.
The converse of this inequality follows from the fact that every blowup of $\mathcal{G}'$ is $\mathcal{F}$-free.
Therefore, we have $\pi(\mathcal{F}) = 6\lambda(\mathcal{G})$.

Now we prove that $\xi(\mathcal{F})=\infty$.
Suppose to the contrary that $\xi(\mathcal{F})=t$ for some $t\in \mathbb{N}^+$.
For every $n \in \mathbb{N}$ let $\mathcal{S}_1(n),\ldots, \mathcal{S}_t(n)$ be the $n$-vertex $3$-graphs that witness
the $t$-stability of $\mathcal{F}$.
For every $i\in [t]$ and $n\in \mathbb{N}$
let $\rho_{i,n} =  \rho(\mathcal{G}',\mathcal{S}_i(n)) = N(\mathcal{G}',\mathcal{S}_i(n))/\binom{n}{m}$ ,
where $N(\mathcal{G}',\mathcal{S}_i(n))$ is the number of copies of $\mathcal{G}'$ in $\mathcal{S}_i(n)$.

We will get a contradiction by first showing that for all sufficiently large $n$
there exists a blowup $\widehat{\mathcal{G}}$ of $\mathcal{G}$ on $n$ vertices whose edge density is close to $\pi(\mathcal{F})$ but
whose shadow density is far from $\rho_{i,n}$ for every $i\in [t]$.
Then we argue that since every shadow edge is contained $\Omega(n)$ edges in $\widehat{\mathcal{G}}$,
the edit-distance of $\widehat{\mathcal{G}}$ and $\mathcal{S}_i(n)$ is $\Omega(n^3)$ for all $i\in [t]$.
This contradicts the definition of $t$-stable.

Notice that $v(\mathcal{G}) = m-2$ if $d_{\mathcal{G}}(v_1, v_2)\ge 2$ and $v(\mathcal{G}^2) = m-2$ if $d_{\mathcal{G}}(v_1, v_2) = 1$.
Let $(z_1,z_2,z_3,\ldots,z_{m-2})\in Z(\mathcal{G})$ (or $Z(\mathcal{G}^2)$ if $d_{\mathcal{G}}(v_1,v_2) = 1$)
be a vector such that $\zeta := \min\{z_1,z_2,z_3,\ldots,z_{m-2}\}$ is maximized.
By Lemma~\ref{LEMMA:max-p-symmetric}, we may assume that $z_1 = z_2$
since otherwise we can replace $z_1$ and $z_2$ by $(z_1+z_2)/2$.

\begin{claim}\label{CLAIM:zeta>0}
We have $\zeta>0$.
\end{claim}
\begin{proof}
If $d(v_1,v_2) \ge 2$, then the minimality of $\mathcal{G}$ implies that
every vector in $Z(\mathcal{G})$ has only positive coordinate.
Therefore, $\zeta > 0$.

Suppose that $d(v_1,v_2) = 1$.
Without loss of generality let us assume that $N_{\mathcal{G}}(v_1,v_2) = \{v_3\}$
and $N_{\mathcal{G}^2}(v_1,v_2) = \{v_3,v_4\}$.
Since $v_4$ is a clone of $v_3$ in $\mathcal{G}^2$,
$(z_1,z_2,z_3,z_4,\ldots,z_{m-2})\in Z(\mathcal{G}^2)$ implies
that $(z_1,z_2,(z_3+z_4)/2,(z_3+z_4)/2,\ldots,z_{m-2})\in Z(\mathcal{G}^2)$.
For the same reason, $(z_1,z_2,z_3,z_4,\ldots,z_{m-2})\in Z(\mathcal{G}^2)$ implies that
$(z_1,z_2,z_3+z_4,z_5,\ldots,z_{m-2})\in Z(\mathcal{G})$.
The minimality of $\mathcal{G}$ implies that each coordinate of $(z_1,z_2,z_3+z_4,z_5,\ldots,z_{m-2})$ is positive.
Therefore, every coordinate in $(z_1,z_2,(z_3+z_4)/2,(z_3+z_4)/2,\ldots,z_{m-2})$ is positive,
and hence, $\zeta > 0$.
\end{proof}

Let $\vec{x}_{\alpha} = \alpha\cdot (z_1,0,0,z_2,z_3,\ldots,z_{m-2})+(1-\alpha)\cdot (0,z_1,z_2,0,z_3,\ldots,z_{m-2})$
for $\alpha \in [0,1]$.
Let $\widehat{\mathcal{G}}_{\alpha}(n) = \mathcal{G}'[V_1,V_1',V_2,V_2',V_3,\cdots,V_{m-2}]$ be a blowup of $\mathcal{G}'$,
where $|V_1| = \lfloor \alpha z_1 n \rfloor$, $|V_1'| = \lfloor (1-\alpha) z_1 n \rfloor$,
$|V_2| = \lfloor \alpha z_2 n \rfloor$, $|V_2'| = \lfloor (1-\alpha) z_2 n\rfloor$,
and $|V_i| = \lfloor z_i n \rfloor$ for $i\in [3,m-2]$.
Let $W = \{v_3,\ldots, v_{m-2}\}$.
Then
\begin{align}
|\partial\widehat{\mathcal{G}}_{\alpha}(n)|
& = \sum_{v_iv_j\in\partial\mathcal{G}\cap\binom{W}{2}}|V_i||V_j|
    + \sum_{v_i\in N_{\mathcal{G}}(v_1)}|V_i|\left(|V_1|+|V_1|'\right)
    + \sum_{v_i\in N_{\mathcal{G}}(v_2)}|V_i|\left(|V_2|+|V_2|'\right) \notag\\
&  \quad +\left(|V_1|+|V_1'|\right)\left(|V_2|+|V_2'|\right)+|V_1||V_1'|+|V_2||V_2'|. \notag
\end{align}
Taking the limit we obtain
\begin{align}
\lim_{n\to \infty}\rho\left(\partial\widehat{\mathcal{G}}_{\alpha}(n)\right)
& = 2\alpha(1-\alpha)(z_1+z_2) + 2C, \notag
\end{align}
where
\begin{align}
C
& = \sum_{v_iv_j\in\partial\mathcal{G}\cap\binom{W}{2}}z_iz_j
    + \sum_{v_i\in N_{\mathcal{G}}(v_1)}z_iz_1
    + \sum_{v_i\in N_{\mathcal{G}}(v_2)}z_iz_2 + z_1z_2 \notag
\end{align}
is independent of $\alpha$.

For $i\in [t+1]$ let $\alpha_i = \frac{i}{2(t+1)}$.
Since $z_1+z_2 > 0$ and $\alpha(1-\alpha)$ is increasing in $\alpha$ for $\alpha \in [0,1/2]$,
there exists $\epsilon>0$ and $N_0$ such that
\begin{align}
\left|\partial\widehat{\mathcal{G}}_{\alpha_j}(n)\right|
        -\left|\partial\widehat{\mathcal{G}}_{\alpha_i}(n)\right|
> \frac{6\epsilon n^2}{\zeta} \notag
\end{align}
for all $1\le i < j \le t+1$.
Fix such an $\epsilon>0$.
By the definition of $t$-stable there exists $\delta= \delta(\epsilon)$ and $N_1 = N_1(\epsilon)$
such that   every $n$-vertex $\mathcal{F}$-free $3$-graph $\mathcal{H}$
with $|\mathcal{H}| \ge (1-\delta)\mathrm{ex}(n,\mathcal{F})$,
satisfies $|\mathcal{H}\triangle \mathcal{S}_i(n)| \le \epsilon n^3$ for some $i\in [t]$.
Since $\lim_{n\to \infty}\rho\left(\widehat{\mathcal{G}}_{\alpha}(n)\right) = \pi(\mathcal{F})$, there exists $N_2$ such that
$\left| \widehat{\mathcal{G}}_{\alpha}(n)\right| \ge (1-\delta)\mathrm{ex}(n,\mathcal{F})$ for all $\alpha\in[0,1]$ and $n\ge N_2$.
Therefore,
for every $j\in [t+1]$ there exists $i\in [t]$ such that
$|\widehat{\mathcal{G}}_{\alpha_j}(n)\triangle\mathcal{S}_i(n)|\le \epsilon n^3$
for all $n\ge \max\{N_0,N_1,N_2\}$.
By the Pigeonhole principle, there exists a pair $\{j,k\}$ such that
$|\widehat{\mathcal{G}}_{\alpha_j}(n)\triangle\mathcal{S}_i(n)|\le \epsilon n^3$
and $|\widehat{\mathcal{G}}_{\alpha_k}(n)\triangle\mathcal{S}_i(n)|\le \epsilon n^3$.
By the triangle inequality, we have $|\widehat{\mathcal{G}}_{\alpha_j}(n)\triangle\widehat{\mathcal{G}}_{\alpha_k}(n)|\le 2\epsilon n^3$.

Without loss of generality we may assume that $\alpha_{j}< \alpha_k$.
Observe that every edge $e\in \partial\widehat{\mathcal{G}}_{\alpha_j}(n)\triangle\partial\widehat{\mathcal{G}}_{\alpha_k}(n)$
is contained in the set $V_1\cup V_1'\cup V_2\cup V_2'$.
Let $\{\ell_1,\ell_2\}\subset [3,m-2]$ be the pair such that $N_{\mathcal{G}^2}=\{\ell_1,\ell_2\}$.
Then we have $N_{\widehat{\mathcal{G}}_{\alpha}(n)}(e) \ge \min\{|V_{\ell_1}|,|V_{\ell_2}|\} \ge \lfloor \zeta n\rfloor> \zeta n/2$.
Therefore, in order to transform $\widehat{\mathcal{G}}_{\alpha_k}(n)$ into $\widehat{\mathcal{G}}_{\alpha_k}(n)$
we need to remove at least
\begin{align}
\left(|\partial\widehat{\mathcal{G}}_{\alpha_k}(n)|-|\partial\widehat{\mathcal{G}}_{\alpha_j}(n)|\right)\frac{\zeta n}{2}
> \frac{6\epsilon n^2}{\zeta} \frac{\zeta n}{2} > 2\epsilon n^3 \notag
\end{align}
edges, a contradiction.
Therefore, $\xi(\mathcal{F}) = \infty$.
\end{proof}
\section{Proof of Theorem~\ref{THM:main-sec1.1}~(a),~(b), and~(c)}\label{SEC:proof-turan-number}
In this section we prove Theorem~\ref{THM:main-sec1.1}~(a),~(b), and~(c).
First let us define $\Gamma_t$ and $\mathcal{F}_t$ that were mentioned in Theorem~\ref{THM:main-sec1.1}.
For convenience, we will use $t+2$ instead $t$ in the rest of this paper.

\begin{definition}
Let $t\ge 1$ be an integer.
\begin{enumerate}[label=(\alph*)]
\item Let
\begin{align}
\Gamma_{t+2} =
\begin{cases}
\{134, 234\}\boxplus \{3,4\} & \text{if } t=1, \\
K_{t+2}^{3}\boxplus \{t+1,t+2\} & \text{if } t\geq 2.
\end{cases} \notag
\end{align}
\item Let $\mathfrak{\Gamma}_{t+2}$ be the collection of all $\Gamma_{t+2}$-colorable $3$-graphs.
\item Let $\gamma_{t+2}(n) = \max\{|\mathcal{H}|\colon v(\mathcal{H}) = n \text{ and } \mathcal{H}\in\mathfrak{\Gamma}_{t+2}\}$.
\item Let $\mathcal{F}_{t+2} = \left\{F\colon v(F)\le 4(t+4)^2 \text{ and } F\not\in \mathfrak{\Gamma}_{t+2}\right\}$.
\end{enumerate}
\end{definition}

\textbf{Remark.}
In the rest of the paper, we always assume that for $t\ge 2$ the vertex set of $\Gamma_{t+2}$ is $[t+4]$, and $t+3, t+4$ are clones of $t+1, t+2$, respectively.


It follows from Proposition~\ref{PROP:nonminimal-corss-blowup} and Lemma~\ref{LEMMA:|H|<=lambda-T-n^r}
that $\gamma_{t+2}(n) \sim \lambda(K_{t+2}^3)n^3 = \frac{t(t+1)}{6(t+2)^2}n^3$
(for $t=1$ we also used $\lambda(\{134,234\}) = \lambda(K_{3}^3)$).

Theorem~\ref{THM:main-sec1.1}~(a) follows easily from Proposition~\ref{PROP:nonminal-finite-family}.
Theorem~\ref{THM:main-sec1.1}~(c) follows easily from Theorem~\ref{THM:nonminimal-general-turan-stability}.
So we just need to prove Theorem~\ref{THM:main-sec1.1}~(b).

\begin{proof}[Proof of Theorem~\ref{THM:main-sec1.1}~(b)]
Let $t\ge 2$ and $n$ be an integer satisfying $(t+2)\mid n$.
It is easy to see that $\mathrm{ex}(n,\mathcal{F}_{t+2}) = \frac{t(t+1)}{6(t+2)^2}n^3$.
Let $\mathcal{H} = \Gamma_{t+2}[V_1,\cdots,V_{t+4}]$ be a blowup of $\Gamma_{t+2}$ with
$|V_1| = \cdots = |V_t| = \frac{n}{t+2}$, $|V_{t+1}| = |V_{t+4}| = \frac{\alpha n}{t+2}$,
and $|V_{t+2}| = |V_{t+3}| = \frac{(1-\alpha) n}{t+2}$, where $\alpha \in [0,1/2]$ satisfies that $\frac{\alpha n}{t+2}\in \mathbb{N}$.
It is easy to see that $|\mathcal{H}| = \frac{t(t+1)}{6(t+2)^2}n^3$.
So it suffices to show that the set $S = \left\{\alpha \in [0,1/2]\colon \frac{\alpha n}{t+2}\in \mathbb{N}\right\}$
has size at least $\frac{n}{2(t+2)}$.
Indeed, suppose that $\alpha = \frac{p}{q}\in [0,1/2]$, where $p\le q/2$ are integers that are coprime.
Notice that $\frac{p}{q}\frac{n}{t+2} \in \mathbb{N}$ if $q\mid \frac{n}{t+2}$.
Since $p$ is coprime with $q$ iff $q-p$ is coprime with $q$,
we have $|\{p\colon (p,q)=1,\ p\le q/2\}| \ge \varphi(q)/2$, where $\varphi(q)$ is the Euler totient function
that denotes the number of positive integers $p\le q$ that are coprime with $q$.
Therefore,  $|S| \ge \frac{1}{2}\sum_{q\mid \frac{n}{t+2}}\varphi(q)$.
It follows from a well known result of Gauss that $\sum_{q\mid \frac{n}{t+2}}\varphi(q) = \frac{n}{t+2}$.
Therefore, $|S| \ge \frac{n}{2(t+2)}$.
This implies that there are at least $\frac{n}{2(t+2)}$ nonisomorphic extremal $\mathcal{F}_t$-free $3$-graphs on $n$ vertices.
The proof for the case $t = 1$ is similar, so we omit it here.
\end{proof}

\section{Stability}\label{SEC:proof-stability}
\subsection{Preparations}\label{SUBSEC:prepare}
In this section we present some useful theorems and lemmas that will be used later
in the proof of Theorem~\ref{THM:main-sec1.1}~(d).

\begin{definition}[Vertex-extendibility]\label{DFN:vertex-extendable}
Let $\mathcal{F}$ be a family of $r$-graphs and let $\mathfrak{H}$ be a class of $\mathcal{F}$-free $r$-graphs.
We say that $\mathcal{F}$ is \emph{vertex-extendable}
with respect to $\mathfrak{H}$ if there exist $\zeta>0$ and $N_0\in\mathbb{N}$ such that
for every $\mathcal{F}$-free $r$-graph $\mathcal{H}$ on $n\ge N_0$
vertices satisfying $\delta(\mathcal{H})\ge \bigl(\pi(\mathcal{F})/(r-1)!-\zeta\bigr)n^{r-1}$
the following holds: if $\mathcal{H}-v$ is a subgraph of a member of $\mathfrak{H}$ for some
vertex $v\in V(\mathcal{H})$, then $\mathcal{H}$ is a subgraph of a member of $\mathfrak{H}$ as well.
\end{definition}

In~\cite{LMR2}, the authors developed a machinery that reduces the proof of stability of certain families $\mathcal{F}$
to the simpler question of checking that an $\mathcal{F}$-free hypergraph $\mathcal{H}$ with large minimum degree is vertex-extendable.



\begin{theorem}[\cite{LMR2}]\label{THM:Psi-trick:G-extendable-implies-degree-stability}
Suppose that $\mathcal{F}$ is a blowup-invariant nondegenerate family of $r$-graphs and that
$\mathfrak{H}$ is a hereditary class of $\mathcal{F}$-free $r$-graphs.
If $\mathfrak{H}$ contains all symmetrized $\mathcal{F}$-free $r$-graphs and
$\mathcal{F}$ is vertex-extendable with respect to $\mathfrak{H}$, then the following statement holds.
There exist $\epsilon>0$ and $N_0$ such that every $\mathcal{F}$-free $r$-graph on $n\ge N_0$ vertices with minimum
degree at least $\left(\pi(\mathcal{F})/(r-1)!-\epsilon\right)n^{r-1}$ is contained in $\mathfrak{H}$.
\end{theorem}

The following lemma will be used extensively in the proof of Theorem~\ref{THM:main-sec1.1}~(d).

\begin{lemma}[see e.g. \cite{LMR1}]\label{LEMMA:greedily-embedding-Gi}
Fix a real $\eta \in (0, 1)$ and integers $m, n\ge 1$.
Let $\mathcal{G}$ be a $3$-graph with vertex set~$[m]$ and let $\mathcal{H}$ be a further $3$-graph
with $v(\mathcal{H})=n$.
Consider a vertex partition $V(\mathcal{H}) = \bigcup_{i\in[m]}V_i$ and the associated
blow-up $\widehat{\mathcal{G}} = \mathcal{G}[V_1,\ldots,V_{m}]$ of $\mathcal{G}$.
If two sets $T \subseteq [m]$ and $S\subseteq \bigcup_{j\not\in T}V_j$
have the properties
\begin{enumerate}[label=(\alph*)]
\item\label{it:47a} $|V_{j}| \ge (|S|+1)|T|\eta^{1/3} n$  for all $j \in T$,
\item\label{it:47b} $|\mathcal{H}[V_{j_1},V_{j_2},V_{j_3}]| \ge |\widehat{\mathcal{G}}[V_{j_1},V_{j_2},V_{j_3}]|
		- \eta n^3$ for all $\{j_1,j_2,j_3\} \in \binom{T}{3}$, and
\item\label{it:47c} $|L_{\mathcal{H}}(v)[V_{j_1},V_{j_2}]| \ge |L_{\widehat{\mathcal{G}}}(v)[V_{j_1},V_{j_2}]|
		- \eta n^2$ for all $v\in S$ and $\{j_1,j_2\} \in \binom{T}{2}$,
\end{enumerate}
then there exists a selection of vertices $u_j\in V_j$ for all $j\in [T]$
such that $U = \{u_j\colon j\in T\}$ satisfies
$\widehat{\mathcal{G}}[U] \subseteq \mathcal{H}[U]$ and
$L_{\widehat{\mathcal{G}}}(v)[U] \subseteq L_{\mathcal{H}}(v)[U]$ for all $v\in S$.
In particular, if $\mathcal{H} \subseteq \widehat{\mathcal{G}}$,
then $\widehat{\mathcal{G}}[U] = \mathcal{H}[U]$ and
$L_{\widehat{\mathcal{G}}}(v)[U] = L_{\mathcal{H}}(v)[U]$ for all $v\in S$.
\end{lemma}

Next we prove some basic properties about $\Gamma_{t+2}$.

\begin{observation}\label{OBS:structure-link}
The following statements hold.
\begin{enumerate}[label=(\alph*)]
\item If a set $U\subset [t+4]$ satisfies that $\Gamma_{t+2}[U]\cong K_{t+2}^3$,
        then either $U = [t]\cup \{t+1,t+4\}$ or $U = [t]\cup \{t+2,t+3\}$.
\item
        We have $L_{\Gamma_{t+2}}(1) \cong \cdots \cong L_{\Gamma_{t+2}}(t)$, and $L_{\Gamma_{t+2}}(t+1) \cong \cdots \cong L_{\Gamma_{t+2}}(t+4).$
\item We have
\begin{align}
d_{\Gamma_{t+2}}(i,j) =
\begin{cases}
t+2 & \text{if } \{i,j\} \in \binom{[t]}{2}, \\
t+1 & \text{if } i\in [t] \text{ and } j\in [t+1,t+4], \\
t   & \text{if } \{i,j\} \in \{\{t+1,t+4\},\{t+2,t+3\}\}, \\
t-1 & \text{if } \{i,j\} \in \{\{t+1,t+3\},\{t+2,t+4\}\}, \\
1   & \text{if } \{i,j\} \in \{\{t+1,t+2\},\{t+3,t+4\}\}.
\end{cases} \notag
\end{align}
\end{enumerate}
\end{observation}

Using Observation~\ref{OBS:structure-link} (a) we prove the following lemma.

\begin{lemma}\label{LEMMA:two-K_t+2}
Suppose that $\mathcal{G}$ is a blowup of $\Gamma_{t+2}$ and $U,U'\subset V(\mathcal{G})$
are two vertex sets of size $t+2$ with $|U\cap U'| = t+1$.
If $\mathcal{G}[U] \cong \mathcal{G}[U'] \cong K_{t+2}^3$,
then the set $U\triangle U'$ is not contained in any edge of $\mathcal{G}$.
\end{lemma}
\begin{proof}
Suppose that $U=\{u_1,\ldots,u_{t+1},u\}$ and $U'=\{u_1,\ldots,u_{t+1},u'\}$.
Let $\phi\colon U\cup U' \to V(\Gamma_{t+2})$ be a map such that $\phi|_{U}$ and $\phi|_{U'}$
are homomorphisms from $\mathcal{G}[U]$ and $\mathcal{G}[U']$ to $\Gamma_{t+2}$, respectively.
Since $|U\cap U'| = t+1$, it follows from Observation~\ref{OBS:structure-link} (a)
that either $\phi(U) = \phi(U') = [t]\cup \{t+1,t+4\}$ or $\phi(U) = \phi(U') = [t]\cup \{t+2,t+3\}$.
In either case, we have $\phi(u) = \phi(u')$, which implies that $u$ and $u'$ lie in the same part of $\mathcal{G}$.
So $\{u,u'\}$ is not contained in any edge of $\mathcal{G}$.
\end{proof}

The following lemma follows easily from Observation~\ref{OBS:structure-link} (c).

\begin{lemma}\label{LEMMA:gamma-t-automorphism}
Suppose that $\phi\colon V(\Gamma_{t+2})\to V(\Gamma_{t+2})$ is a homomorphism from $\Gamma_{t+2}$ to $\Gamma_{t+2}$.
Then $\phi$ is bijective,
$\phi([t]) = [t]$, and $$\left\{ \phi(\{t+1,t+4\}),\phi(\{t+2,t+3\}) \right\} = \left\{ \{t+1,t+4\},\{t+2,t+3\} \right\}.$$
If $t\ge 3$, then additionally, $\phi([t-1]) = [t-1]$.
\end{lemma}

Using Observation~\ref{OBS:structure-link} (c) we further prove the following lemma.

\begin{lemma}\label{LEMMA:gamma-t-partial-embedding}
Suppose that $\phi\colon V(\Gamma_{t+2})\setminus \{t+3\} \to V(\Gamma_{t+2})$
is a homomorphism from $\Gamma_{t+2}\setminus \{t+3\}$ to $\Gamma_{t+2}$.
Then $\phi$ is injective, $\phi([t]) = [t]$, $\phi(\{t+1,t+2,t+4\})\subset [t+1,t+4]$,
and $\{\phi(t+1),\phi(t+4)\}\in \{\{t+1,t+4\},\{t+2,t+3\}\}$.
If $t\ge 3$, then additionally, $\phi([t-1]) = [t-1]$.
\end{lemma}
\begin{proof}
Let $F = \Gamma_{t+2}\setminus\{t+3\}$.
Since $\phi$ is an embedding, we have $d_{F}(i,j) \le d_{\Gamma_{t+2}}(\phi(i),\phi(j))$ for all pairs of vertices in $F$.
Observe that
\begin{align}
d_{F}(i,j) =
\begin{cases}
t+1 & \text{if } \{i,j\} \in \binom{[t]}{2} \text{ or } (i,j)\in [t-1]\times\{t+4\} \text{ or } (i,j) = (t,t+1), \\
t & \text{if } (i,j)\in [t-1] \times \{t+1,t+2\} \text{ or } (i,j)\in \{t\}\times \{t+2,t+4\}\\
 & \text{or } \{i,j\} =\{t+1,t+4\}, \\
t-1 & \text{if } \{i,j\} =\{t+2,t+4\}, \\
1   & \text{if } \{i,j\} =\{t+1,t+2\}.
\end{cases} \notag
\end{align}
Since the induced subgraph of $F$ on $[t]\cup \{t+1,t+4\}$ is isomorphic to $K_{t+2}^{3}$,
it follows from from Observation~\ref{OBS:structure-link} (a) that
$\phi([t]\cup \{t+1,t+4\}) \in \{[t]\cup \{t+1,t+4\},[t]\cup \{t+2,t+3\}\}$.
By symmetry we may assume that $\phi([t]\cup \{t+1,t+4\}) = [t]\cup \{t+1,t+4\}$.
Then we have $\phi(t+2) \in [t+1,t+4]\setminus \phi([t]\cup \{t+1,t+4\}) = \{t+2,t+3\}$
By symmetry we may assume that $\phi(t+2) = t+2$.
Since $d_{\Gamma_{t+2}}(\phi(t+2),t+1) = 1$ and $d_{\Gamma_{t+2}}(\phi(t+2),t+4) = t-1$,
we have $\phi(t+1) = t+1$ and $\phi(t+4) = t+4$
(otherwise we would have $\phi(i)\in \{t+1,t+4\}$ for some $i\in [t]$,
but this implies that $t = d_{F}(i,t+2) \le d_{\Gamma_{t+2}}(\phi(i),\phi(t+2)) \le t-1$, a contradiction).
Consequently, $\phi([t]) = [t]$.

Suppose that $t\ge 3$.
Let $S = \{t+1,t+2,t+4\}$
Observe that $L_{F}(i)[S] = L_{F}(j)[S] \neq L_{F}(t)(S)$ and
$L_{\Gamma_{t+2}}(i)[\phi(S)] = L_{\Gamma_{t+2}}(j)[\phi(S)] \neq L_{\Gamma_{t+2}}(t)(\phi(S))$ and for all $i,j\in [t-1]$.
So, we should have $\phi([t-1]) = [t-1]$ and $\phi(t) = \phi(t)$.
\end{proof}

The following lemma characterizes the vectors $(x_1,\ldots, x_{t+4}) \in \Delta_{t+3}$ for which
the value $p_{\Gamma_{t+2}}(x_1,\ldots, x_{t+4})$ is close to the maximum.

\begin{lemma}\label{LEMMA:Lagrangian-stability}
For every integer $t\ge 2$ and every real number $\epsilon>0$ there exists a constant $\delta>0$ such that
if a vector $\vec{x}\in \Delta_{t+3}$ satisfies $p_{\Gamma_{t+2}}(x_1,\ldots, x_{t+4}) \ge \lambda(K_{t+2}^3) - \delta$,
then there exists a real number $\alpha \in [0,1]$ such that
\begin{align}\label{equ:weight-stable}
x_i =
\begin{cases}
\frac{1}{t+2} \pm \epsilon & \text{if } i\in [t], \\
\frac{\alpha}{t+2} \pm \epsilon & \text{if } i\in \{t+1, t+4\},\\
\frac{1-\alpha}{t+2} \pm \epsilon & \text{if } i\in\{t+2, t+3\}.
\end{cases}
\end{align}
\end{lemma}
\begin{proof}
We will use
\begin{align}\label{INEQU:Maclaurin-r=2}
xy = \left(\frac{x+y}{2}\right)^2 -  \left(\frac{x-y}{2}\right)^2,
\end{align}
and the following inequalities (see Lemma 2.2 in~\cite{LMR2}),
\begin{align}\label{INEQU:Maclaurin-r=3}
p_{K_{t+2}^3}(y_1,\ldots,y_{t+2})
\le \frac{t(t+1)}{2(t+2)^2}- \frac{t}{6(t+2)}\sum_{i=1}^{t+2}\left(y_i-\frac{1}{t+2}\right)^2
\end{align}
for all $(y_1,\ldots, y_{t+2})\in \Delta_{t+1}$.

Let $\delta = \left(\frac{\epsilon}{30t}\right)^2$ and note that $\epsilon=30t\delta^{1/2}$.
Let $p = p_{\Gamma_{t+2}}(x_1,\ldots, x_{t+4})$ and $\sigma_{i} = \sigma_i(x_1,\ldots, x_t) =\sum_{E\in \binom{[t]}{i}}\prod_{j\in E}x_j$
for $i\in \{1,2,3\}$.
Let $y_{12} = x_{t+1}+x_{t+2}$, $y_{34} = x_{t+3}+x_{t+4}$, $y_{13} = x_{t+1}+x_{t+3}$, $y_{24} = x_{t+2}+x_{t+4}$,
and $y_{t+1} = y_{t+2} = (x_{t+1}+x_{t+2}+x_{t+3}+x_{t+4})/2$.
Note that $y_{t+1} = y_{t+2} = (y_{12}+y_{34})/2 = (y_{13}+y_{24})/2$.
By Equation $(\ref{INEQU:Maclaurin-r=2})$, we have
\begin{align}
p
& = \sigma_{3} + \sigma_{2}\left(1-\sigma_1\right)+\sum_{i=1}^{t-1}x_iy_{12}y_{34} + x_{t}y_{13}y_{24} \notag\\
& \le \sigma_{3} + \sigma_{2}\left(1-\sigma_1\right) + \sigma_{1}y_{t+1}y_{t+2}
    - (\sigma_1-x_{t})\left(\frac{y_{12}-y_{34}}{2}\right)^2 - x_{t}\left(\frac{y_{13}-y_{24}}{2}\right)^2 \notag\\
& = p_{K_{t+2}^{3}}(x_1,\ldots,x_{t},y_{t+1},y_{t+2})
        - (\sigma_1-x_{t})\left(\frac{y_{12}-y_{34}}{2}\right)^2 - x_{t}\left(\frac{y_{13}-y_{24}}{2}\right)^2. \notag
\end{align}
Since $p \ge \frac{t(t+1)}{2(t+2)^2} - \delta$, it follows from Equation $(\ref{INEQU:Maclaurin-r=3})$
and the inequality above that
\begin{align}\label{equ:delta}
\delta
& \ge \frac{t}{6(t+2)}\sum_{i=1}^{t}\left(x_i-\frac{1}{t+2}\right)^2+\sum_{i=t+1}^{t+2}\left(y_i-\frac{1}{t+2}\right)^2 \notag\\
& + (\sigma_1-x_{t})\left(\frac{y_{12}-y_{34}}{2}\right)^2+x_{t}\left(\frac{y_{13}-y_{24}}{2}\right)^2.
\end{align}
First, it follows that
\begin{align}
\left|x_i-\frac{1}{t+2}\right| & \le\sqrt{\frac{6(t+2)}{t}\delta}\le 6\delta^{1/2} \quad\text{for }i\in [t],\text{ and}\notag\\
\left|y_i-\frac{1}{t+2}\right| & \le\sqrt{\frac{6(t+2)}{t}\delta}\le 6\delta^{1/2} \quad\text{for }i\in \{t+1,t+2\}. \notag
\end{align}
In particular, the first inequality implies that $x_t \ge 1/(t+2)- 6\delta^{1/2} \ge 1/4t^2$ and  $\sigma_1 -x_t = \sum_{i=1}^{t-1}x_i \ge 1/4t^2$.
So it follows from Inequality $(\ref{equ:delta})$ implies that
$$\max\{|y_{12}-y_{34}|,|y_{13}-y_{24}|\}\le \max\left\{2\sqrt{\delta/x_{t}},2\sqrt{\delta/(\sigma_1-x_{t})} \right\} \le 4t\delta^{1/2}.$$
Since $y_{12}+y_{34} = y_{13}+y_{24} = 2y_{t+1}$, we obtain
\begin{align}
\max\left\{\left|y_{12}-\frac{1}{t+2}\right|,\left|y_{23}-\frac{1}{t+2}\right|,
\left|y_{13}-\frac{1}{t+2}\right|,\left|y_{24}-\frac{1}{t+2}\right|\right\}
\le 4t\delta^{1/2}+6\delta^{1/2} \le 10t\delta^{1/2}. \notag
\end{align}
Consequently,
\begin{align}
|x_{t+1}-x_{t+4}| & = |y_{12}-y_{24}| \le 20t\delta^{1/2}, \quad\text{and}\quad
|x_{t+2}-x_{t+3}| & = |y_{12}-y_{13}| \le 20t\delta^{1/2}. \notag
\end{align}
Since $x_{t+1}+x_{t+2} = y_{12} = \frac{1}{t+2}\pm 10t\delta^{1/2}$,
there exists $\alpha \in [0,1]$ such that
\begin{align}
x_{t+1} = \frac{\alpha}{t+2}\pm 10t\delta^{1/2}, \quad\text{and}\quad
x_{t+2} = \frac{1-\alpha}{t+2}\pm 10t\delta^{1/2}. \notag
\end{align}
Consequently,
\begin{align}
x_{t+3} = \frac{1-\alpha}{t+2}\pm 30t\delta^{1/2}, \quad\text{and}\quad
x_{t+4} = \frac{\alpha}{t+2}\pm 30t\delta^{1/2}. \notag
\end{align}
This proves Lemma~\ref{LEMMA:Lagrangian-stability}.
\end{proof

\subsection{Proof of Theorem~\ref{THM:main-sec1.1}~(d)}\label{SUBSEC:proof-stability}
We prove Theorem~\ref{THM:main-sec1.1}~(d) in this section.
Recall from Lemma~\ref{LEMMA:F-free-equ-F-homo-free} that $\mathcal{F}_{t+2}$ is blowup-invariant,
and from Claim~\ref{CLAIM:symmetrized-stable} that the hereditary family $\mathfrak{\Gamma}_{t+2}$ contains all symmetrized $\mathcal{F}_{t+2}$-free $3$-graphs.
Therefore, by Theorem~\ref{THM:Psi-trick:G-extendable-implies-degree-stability}, it suffices to show that $\mathcal{F}_{t+2}$ is vertex-extendable with respect to $\mathfrak{\Gamma}_{t+2}$.

\begin{proof}[Proof of Theorem~\ref{THM:main-sec1.1}~(d)]
Let $\epsilon>0$ be a sufficiently small constant and $n$ be a sufficiently large integer.
Let $\delta>0$ be the constant guaranteed by Lemma~\ref{LEMMA:Lagrangian-stability}.
We may assume that $\delta \le \epsilon$.
Let $\mathcal{H}$ be an $(n+1)$-vertex $\Gamma_{t+2}$-colorable $3$-graph with minimum degree at least
$3(\lambda - \delta/2)(n+1)^2$, where $\lambda = \lambda(\Gamma_{t+2}) = \frac{t(t+1)}{6(t+2)^2}$.

Suppose that there exists a vertex $v\in V(\mathcal{H})$ such that $\mathcal{H}\setminus\{v\} \in \mathfrak{\Gamma}_{t+2}$.
Let $V = V(\mathcal{H})\setminus \{v\}$ and $\mathcal{H}' = \mathcal{H}\setminus \{v\}$.
Let $V = V_1\cup \cdots \cup V_{t+4}$ be a partition
such that the map $\phi\colon V(\mathcal{H}') \to V(\Gamma_{t+2})$
defined by $\phi(V_i) = i$ for $i\in [t+4]$ induces a homomorphism from $\mathcal{H}'$ to $\Gamma_{t+2}$.
Let $\mathcal{G} = \Gamma_{t+2}[V_1,\ldots,V_{t+4}]$ be a blowup of $\Gamma_{t+2}$.
It is possible that some set $V_i$ is empty. So it will be convenient later to define the following graphs.

Let $w_1,\ldots, w_{t+4}$ be $t+4$ new vertices and let $W_i = V_i\cup \{w_i\}$ for $i\in [t+4]$.
Let $\widehat{\mathcal{G}} = \Gamma_{t+2}[W_1,\ldots,W_{t+4}]$ be a blowup of $\Gamma_{t+2}$.
For every $i\in [t+4]$ define $L_{\mathcal{G}}(i) = L_{\widehat{\mathcal{G}}}(w_i)[V]$
and $N_{\mathcal{G}}(i) = V\setminus V_i$.
Notice that $L_{\mathcal{G}}(i)$ is a graph on $N_{\mathcal{G}}(i)$.

We will $N(u)$ and $L(u)$ to denote $N_{\mathcal{H}}(u)$ and $L_{\mathcal{H}}(u)$ respectively
for every vertex $u\in V(\mathcal{H})$ if it does not cause any confusion.

\begin{claim}\label{CLAIM:stability-proof-goal}
It suffices to prove that there exists some $i_0\in [t+4]$ such that
$L(v)\subset L_{\mathcal{G}}(i_0)$.
\end{claim}
\begin{proof}
Suppose that there exists some $i_0\in [t+4]$ such that
$L(v)\subset L_{\mathcal{G}}(i_0)$.
First notice that since $N_{\mathcal{G}}(i_0)\cap V_{i_0} = \emptyset$, we have $N(v)\cap V_{i_0} = \emptyset$ as well.
Let $\widehat{V}_{i_0} = V_{i_0}\cup\{v\}$ and $\widehat{V}_i = V_i$ for $i\in [t+4]\setminus \{i_0\}$.
It is clear that $\mathcal{H}\subset \Gamma_{t+2}[\widehat{V}_1,\ldots,\widehat{V}_{t+4}]$.
Therefore, $\mathcal{F}_{t+2}$ is vertex-extendable with respect to $\mathfrak{\Gamma_{t+2}}$.
This proves Theorem~\ref{THM:main-sec1.1}~(d).
\end{proof}

Let $x_i = |V_i|/n$ for $i\in [t+4]$.
By assumption, we have
$$\delta(\mathcal{H}') \ge \delta(\mathcal{H})-n \ge 3(\lambda - \delta/2)(n+1)^2-n \ge 3(\lambda - \delta)n^2,$$
and hence $|\mathcal{H}'| \ge \delta(\mathcal{H}')n/3 \ge (\lambda-\delta)n^3$.
Since $p_{\Gamma_{t+2}}(x_1,\ldots,x_{t+4}) \ge |\mathcal{H}'|/n^3 \ge \lambda-\delta$,
it follows from Lemma~\ref{LEMMA:Lagrangian-stability} that
there exists a real number $\alpha \in [0,1]$ such that Equation $(\ref{equ:weight-stable})$ holds.


\begin{claim}\label{CLAIM:missing-link-few}
For every vertex $u\in V$ we have
$|L_{\mathcal{G}}(u)\setminus L(u)| \le 4t \epsilon n^2$.
\end{claim}
\begin{proof}
Let us assume that $u\in V_i$ where $i\in [t+4]$.

If $i\in \{1,\ldots, t\}$, then it follows from $(\ref{equ:weight-stable})$ that
\begin{align}
|L_{\mathcal{G}}(u)|/n^2
& \le \binom{t-1}{2}\left(\frac{1}{t+2} + \epsilon\right)^2
    + (t-1)\left(\frac{1}{t+2} + \epsilon\right)\left(\frac{2}{t+2} + 4\epsilon\right) \notag\\
&  \quad  + \left(\frac{1}{t+2} + 2\epsilon\right)\left(\frac{1}{t+2} + 2\epsilon \right) \notag\\
& \le \frac{t(t+1)}{2(t+2)^2} + \frac{t(t+3)}{t+2}\epsilon + \frac{t^2+5t+2}{2}\epsilon^2
 \le \frac{t(t+1)}{2(t+2)^2} + 2t\epsilon. \notag
\end{align}

If $i \in \{t+1,\ldots, t+4\}$,
then it follows from $(\ref{equ:weight-stable})$ that
\begin{align}
|L_{\mathcal{G}}(u)|/n^2
& \le \binom{t}{2}\left(\frac{1}{t+2} + \epsilon\right)^2
    + t\left(\frac{1}{t+2} + \epsilon\right)\left(\frac{1}{t+2} + 2\epsilon\right) \notag\\
& \le \frac{t(t+1)}{2(t+2)^2} + t\epsilon + \frac{t(t+3)}{2}\epsilon^2
\le \frac{t(t+1)}{2(t+2)^2} + 2t\epsilon. \notag
\end{align}

Recall that $\lambda = \frac{t(t+1)}{6(t+2)^2}$ and $\delta\le \epsilon$.
Therefore,
\begin{align}
|L_{\mathcal{G}}(u)\setminus L(u)|/n^2
\le \frac{t(t+1)}{2(t+2)^2} + 2t\epsilon - 3\left(\lambda-\delta\right)
\le \frac{t(t+1)}{2(t+2)^2} + 2t\epsilon - \frac{t(t+1)}{2(t+2)^2} + 3\epsilon
\le
4t\epsilon. \notag
\end{align}
\end{proof


\begin{claim}\label{CLAIM:missing-neighbors-small}
For every $i\in [t+4]$ and every  $u\in V_i$ we have $|V_j\setminus N(u)| \le 12t\epsilon n$ for all $j\in [t+4]\setminus\{i\}$.
\end{claim}
\begin{proof}
Fix $i\in [t+4]$ and let $u\in V_i$.
If $i \in [t]$, then it follows from Lemma~\ref{LEMMA:Lagrangian-stability} that
\begin{align}
\delta(L_{\mathcal{G}}(u))/n
& \ge \min\left\{(t-2)\left(\frac{1}{t+2}-\epsilon\right)
        +2\left(\frac{\alpha}{t+2}-\epsilon+\frac{1-\alpha}{t+2}-\epsilon\right),\right. \notag\\
&  \quad  \left. (t-1)\left(\frac{1}{t+2}-\epsilon\right)+\frac{\alpha}{t+2}-\epsilon+\frac{1-\alpha}{t+2}-\epsilon\right\}
 \ge \frac{t}{t+2}-(t+2)\epsilon
 \ge \frac{1}{3}. \notag
\end{align}
If $i \in [t+1,t+4]$, then it follows from Lemma~\ref{LEMMA:Lagrangian-stability} that
\begin{align}
\delta(L_{\mathcal{G}}(u))/n
& \ge \min\left\{(t-1)\left(\frac{1}{t+2}-\epsilon\right)
        +\frac{\alpha}{t+2}-\epsilon+\frac{1-\alpha}{t+2}-\epsilon, t\left(\frac{1}{t+2}-\epsilon\right)\right\} \notag\\
& \ge \frac{t}{t+2}-(t+2)\epsilon
\ge \frac{1}{3}. \notag
\end{align}
Since each vertex in $V_j\setminus N(u)$ contributes at least $\delta(L_{\Gamma_{t+2}}(u))$ edges to
$L_{\mathcal{G}}(u)\setminus L(u)$,
it follows from Claim~\ref{CLAIM:missing-link-few} that
\begin{align}
|V_j\setminus N(u)|
\le \frac{4t\epsilon n^2}{n/3} = 12t\epsilon n. \notag
\end{align}
\end{proof}

By symmetry, we may assume that $\alpha \ge 1/2$.
Since $\epsilon$ is sufficiently small, we have $\min\{x_1,\ldots, x_t, x_{t+1}, x_{t+4}\} \ge \frac{1}{4(t+2)}$.

\begin{claim}\label{CLAIM:link-empty-inside}
We have $L(v) \cap \binom{V_i}{2} = \emptyset$ for all $i\in [t+4]$.
\end{claim}
\begin{proof}
Suppose to the contrary that there exists $i\in [t+4]$ such that $L(v) \cap \binom{V_i}{2} \neq \emptyset$.
Let us assume that $\{u_i,u_i'\} \in L(v) \cap \binom{V_i}{2}$.

First let us assume that  $i\in \{1,\ldots, t,t+1,t+4\}$.
Applying Lemma~\ref{LEMMA:greedily-embedding-Gi} with $\eta = 4t\epsilon$
and $T = \{1,\ldots, t,t+1,t+4\}\setminus \{i\}$
we obtain $u_j \in V_j$ for every $j\in T$ such that sets $U = \{u_j\colon j\in T\}\cup \{u_i\}$
and $U' = \{u_j\colon j\in T\}\cup \{u_i'\}$
satisfy
$$\mathcal{H}[U] = \mathcal{G}[U] \cong K_{t+2}^{3}\quad\text{and}\quad
    \mathcal{H}[U'] = \mathcal{G}[U']\cong K_{t+2}^{3}.$$

Let $F = \mathcal{H}[U\cup U']\cup \{\{v,u_i,u_i'\}\}$.
It is clear that $v(F) = t+4\le 4(t+4)^2$.
Therefore, $F$ is contained as a subgraph in some blowup of $\Gamma_{t+2}$.
Since $\mathcal{H}[U]\cong K_{t+2}^{3}$, $\mathcal{H}[U']\cong K_{t+2}^{3}$, and $|U\cap U'| = t+1$,
it follows from Lemma~\ref{LEMMA:two-K_t+2} that $\{u_i,u_i'\}$ is not contained in any edge of $F$, a contradiction.

Now let us assume that $i\in \{t+2,t+3\}$.
Applying Lemma~\ref{LEMMA:greedily-embedding-Gi} with $\eta = 4t\epsilon$ and $T = \{1,\ldots, t,t+1,t+4\}$
we obtain $u_j \in V_j$ for every $j\in T$ such that the set $U = \{u_j\colon j\in T\}$ satisfies
\begin{align}
\mathcal{H}[U] = \mathcal{G}[U] \cong K_{t+2}^{3}\quad\text{and}\quad
L(u_i)[U]  = L_{\mathcal{G}}(u_i)[U] = L_{\mathcal{G}}(u_i')[U]= L(u_i')[U]. \notag
\end{align}

Let $F = \mathcal{H}[U\cup \{u_i,u_i'\}]\cup \{\{v,u_i,u_i'\}\}$.
It is clear that $v(F) = t+5 \le 4(t+4)^2$.
So there exists a homomorphism $\phi\colon V(F)\to V(\Gamma_{t+2})$ from $F$ to $\Gamma_{t+2}$.
Since $F[U]\cong K_{t+2}^{3}$, it follows from Observation~\ref{OBS:structure-link} (a) that either
$\phi(U) = [t]\cup \{t+1,t+4\}$ or $\phi(U) = [t]\cup \{t+2,t+3\}$.
By symmetry we may assume that $\phi(U) = [t]\cup \{t+1,t+4\}$.
Since $U\cup \{u_i,u_i'\}$ is $2$-covered in $F$,
we have $\{\phi(u_i),\phi(u_i')\} = [t+4]\setminus \phi(U) = \{t+2,t+3\}$.
However, notice that $|L_{F}(u_i)[U] \cap L_{F}(u_i')[U]| = \binom{t}{2}+t$,
while $|L_{\Gamma_{t+2}}(t+2)[\phi(U)] \cap L_{\Gamma_{t+2}}(t+3)[\phi(U)]| = \binom{t}{2}<\binom{t}{2}+t$, a contradiction.
\end{proof


Define
\begin{align}
I_{small}  = \left\{i\in [t+4]\colon |N(v)\cap V_i|\le \epsilon^{1/4}n\right\}, \quad\text{and}\quad
I_{\emptyset}  = \left\{i\in [t+4]\colon N(v)\cap V_i = \emptyset\right\}. \notag
\end{align}
It follows from the definition that $I_{\emptyset} \subset I_{small}$.

Next we will consider two cases depending on the sizes of $V_{t+2}$ and $V_{t+3}$:
either $V_{t+2}$ and $V_{t+3}$ are both large (i.e. at least $50t^2\epsilon^{1/4}$) or $V_{t+2}$ and $V_{t+3}$ are both small.
The first case is quite easy to handle while the second case needs more work.

\bigskip

\textbf{Case 1}: $\alpha \le 1 - 101t^3\epsilon^{1/4}$.

Note that in this case we have
$$\min\left\{|V_{t+1}|,|V_{t+2}|,|V_{t+3}|,|V_{t+4}|\right\}\ge \frac{101t^3\epsilon^{1/4}}{t+2}n \ge 50t^2\epsilon^{1/4}n.$$

Define an auxiliary graph $R$ on $[t+4]$ in which two vertices $i$ and $j$ are adjacent
iff there exists $e\in L(v)$ such that $e\cap V_{i} \neq \emptyset$ and $e\cap V_{j} \neq \emptyset$.
An easy observation is that $L(v)\subset R[V_1,\ldots,V_{t+4}]$.


\begin{claim}\label{CALIM:case1-R1-structure}
We have $L(v) \subset L_{\mathcal{G}}(i)$ for some $i\in I_{small}$.
\end{claim}
\begin{proof}
For each edge $ij \in R$ let $e_{ij} \in L(v)$ be an edge
with one endpoint in $V_i$ and the other endpoint in $V_j$.
Let $U_1 = \bigcup_{ij\in R}e_{ij}$ be a vertex subset of $V$ and note that $|U_1| \le 2\binom{t+4}{2}$.
Define
\begin{align}
V_i' =
\begin{cases}
V_i \cap N(v) \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right) & \text{for } i \in [t+4]\setminus I_{small}, \\
V_i \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right) & \text{for } i \in I_{small}.
\end{cases} \notag
\end{align}
It follows from Claim~\ref{CLAIM:missing-neighbors-small} and our assumption that
\begin{align}
|V_i'|\ge \min\left\{\epsilon^{1/4}n - \left(|U_1|+1\right)\times 12t\epsilon n,\
                50t^2\epsilon^{1/4}n - |U_1|\times 12t\epsilon n\right\} \ge 4(t+4)^3\epsilon^{1/3}n. \notag
\end{align}
Applying Lemma~\ref{LEMMA:greedily-embedding-Gi} with $\eta = 4t\epsilon$ and $T = [t+4]$
we obtain $u_i \in V_i'$ for $i\in T$ such that the set $U_2 = \{u_i\colon i\in T\}$ satisfies
\begin{enumerate}[label=(\alph*)]
  \item $\mathcal{H}[U_2] = \mathcal{G}[U_2] \cong \Gamma_{t+2}$, and
  \item $L(u)[U_2] = L_{\mathcal{G}}(u)[U_2]$ for all $u\in U_1$.
\end{enumerate}
Let $F = \mathcal{H}[U_1\cup U_2 \cup \{v\}]$.
Note that $v(F) \le 2\binom{t+4}{2}+ t+5\le  3(t+4)^2$.
By assumption there exists a homomorphism $\phi\colon V(F) \to V(\Gamma_{t+2})$ from $F$ to $\Gamma_{t+2}$
(otherwise we would have $F\in\mathcal{F}_{t+2}$, a contradiction).
Since $F[U_2] \cong \Gamma_{t+2}$, it follows from Lemma~\ref{LEMMA:gamma-t-automorphism} that
$\phi(u_{i})\in [t]$ for $i\in [t]$ (and $\phi(u_t) = t$ if $t\ge 3$), and
\begin{align}
\left\{ \phi(\{u_{t+1},u_{t+4}\}),\phi(\{u_{t+2},u_{t+3}\}) \right\} = \left\{ \{t+1,t+4\},\{t+2,t+3\} \right\}. \notag
\end{align}
By symmetry we may assume that $\phi(u_i) = i$ for $i \in [t+4]$.
For every $i\in [t+4]$ and $u\in V_i\cap U_1$ since $u$ is adjacent to all vertices in $U_2\setminus \{u_i\}$ in $\partial F$,
we have $\phi(u) = i$.
Finally, since $v$ is adjacent to all vertices in $\{u_i\colon i\in [t+4]\setminus I_{small}\}$,
we have $\phi(v)\not\in \phi([t+4]\setminus I_{small})$.
In other words, $\phi(v)\in \phi(I_{small})$.
This means that there exists some $i\in I_{small}$ such that
$\phi(R)\subset L_{\Gamma_{t+2}}(\phi(i))$, and hence $L(v) \subset L_{\mathcal{G}}(i)$.
\end{proof


\medskip

\textbf{Case 2}: $\alpha \ge 1 - 101t^3\epsilon^{1/4}$.

Note that in this case we have
\begin{align}
\max\left\{|V_{t+2}|,|V_{t+3}|\right\} & \le \frac{101t^3\epsilon^{1/4}}{t+2}n+\epsilon n \le 102t^2\epsilon^{1/4}n, \quad\text{and}\notag\\
\min\left\{|V_{t+1}|,|V_{t+4}|\right\} & \ge \frac{1-100t^3\epsilon^{1/4}}{t+2}n-\epsilon n \ge \frac{n}{t+2}-50t^2\epsilon^{1/4}n. \notag
\end{align}

For convenience, let $S = [t]\cup \{t+1,t+4\}$.

\begin{claim}\label{CLAIM:case2-at-most-one-small-intersection}
We have $|I_{small}\cap S| \le 1$.
\end{claim}
\begin{proof}
Suppose to the contrary that $|I_{small}\cap S| \ge 2$.
Then it follows from Lemma~\ref{LEMMA:Lagrangian-stability} that
\begin{align}
|L(v)|/n^2
& \le \binom{t}{2}\left(\frac{1}{t+2}+\epsilon\right)^2 + 2\times 102t^2\epsilon^{1/4} + 2\epsilon^{1/4} \notag\\
& =    \frac{t(t+1)}{2(t+2)^2} - \frac{t}{(t+2)^2} +\frac{t(t-1)}{t+2}\epsilon
    +  \frac{t(t-1)}{2}\epsilon^2+204t^2\epsilon^{1/4} + 2\epsilon^{1/4} \notag\\
& < 3(\lambda-\delta), \notag
\end{align}
a contradiction.
\end{proof

Next we will consider two cases depending on the value of $|I_{small}\cap S|$: either $|I_{small}\cap S| = 1$ or $|I_{small}\cap S| = 0$.

\textbf{Case 2.1}: $|I_{small}\cap S| = 1$.

We may assume that $I_{small} = \{1\}$. The proof for other cases follows analogously.
Let $W = \bigcup_{i\in S\setminus I_{small}}V_i$.
It follows from our assumption that
\begin{align}
|L(v)[W]|/n^2
\ge \delta(\mathcal{H})/n^2 - 2\times 102t^2\epsilon^{1/4} - \epsilon^{1/4}
\ge \frac{t(t+1)}{2(t+2)^2} - 205t^2\epsilon^{1/4}. \notag
\end{align}
Notice that the induced subgraph of $L_{\mathcal{G}}(1)$ on $W$
is the blowup $K_{t+1}[V_2,\ldots,V_{t},V_{t+1},V_{t+4}]$ of $K_{t+1}$.
Using the same argument as in the proof of Claim~\ref{CLAIM:missing-link-few}, we get
\begin{align}
|L_{\mathcal{G}}(1)\setminus L(v)[W]|/n^2
\le \frac{t(t+1)}{2(t+2)^2} + 2t\epsilon - \left(\frac{t(t+1)}{2(t+2)^2} - 205t^2\epsilon^{1/4}\right)
\le 206t^2\epsilon^{1/4}. \notag
\end{align}
Also, using the same argument as in the proof of Claim~\ref{CLAIM:missing-neighbors-small}, we get
\begin{align}
|N(v)\cap V_{i}| \ge |V_i| - \frac{206t^2\epsilon^{1/4}n^2}{n/3}
        \ge \frac{n}{t+2} -50 t^2\epsilon^{1/4}n - 700t^2\epsilon^{1/4}n
        \ge \frac{n}{2(t+2)}. \notag
\end{align}


\begin{claim}\label{CLAIM:case2-no-neighbor-in-V1}
We have $N(v)\cap V_1 = \emptyset$.
\end{claim}
\begin{proof}
Suppose to the contrary that there exists a vertex $u_1\in N(v)\cap V_1$.
Let $V_i' = V_i\cap N(v) \cap N(u_1)$ for $i\in S\setminus \{1\}$.
Notice that $|V_i'| \ge \frac{n}{2(t+2)} - 12t\epsilon n > 206(t+4)^4\epsilon^{1/12} n$.
Applying Lemma~\ref{LEMMA:Lagrangian-stability} with $\eta = 206t^2\epsilon^{1/4}$ and $T = S\setminus \{1\}$,
we obtain a vertex $u_i \in V_i'$ for every $i\in T$ such that the set $U = \{u_i\colon i\in T\}$ satisfies
\begin{align}
\mathcal{H}[U\cup \{u_1\}] = \mathcal{G}[U\cup \{u_1\}] \cong K_{t+2}^{3}
\quad\text{and}\quad
\mathcal{H}[U\cup \{v\}] = \mathcal{G}[U\cup \{v\}] \cong K_{t+2}^{3}. \notag
\end{align}
Let $e\in \mathcal{H}$ be an edge that contains $\{v,u_1\}$ and let $F = \mathcal{H}[U\cup\{v,u_1\}]\cup\{e\}$.
By assumption, $F$ should be contained in some blowup of $\Gamma_{t+2}$,
but this would contradict Lemma~\ref{LEMMA:two-K_t+2} since $\{v,u_1\}$ is contained in $e$.
\end{proof

\begin{claim}\label{CLAIM:case2-put-v-in-V1}
We have $L(v) \subset L_{\mathcal{G}}(1)$.
\end{claim}
\begin{proof}
Suppose to the contrary that there exists an edge $e\in L_{\mathcal{G}}(1)\setminus L(v)$.
By symmetry we may assume that $e = \{u_{t+1},u_{t+2}\}$ and $(u_{t+1},u_{t+2})\in V_{t+1}\times V_{t+2}$.
Let
\begin{align}
V_{i}' =
\begin{cases}
V_{i}\cap N(u_{t+1}) \cap N(u_{t+2}) & \text{if } i = 1, \\
V_{i}\cap N(v) \cap N(u_{t+2}) & \text{if } i = t+1, \\
V_{i}\cap N(v) \cap N(u_{t+1}) \cap N(u_{t+2}) & \text{if } i\in [2,t]\cup \{t+4\}.
\end{cases}\notag
\end{align}
Notice that $|V_i'| \ge \frac{n}{2(t+2)} - 2\times 12t\epsilon > 206(t+4)^4\epsilon^{1/12}$.
Applying Lemma~\ref{LEMMA:Lagrangian-stability} with $\eta = 206t^2\epsilon^{1/4}$ and $T = S$,
we obtain a vertex $u_i' \in V_i'$ for every $i\in T$ such that the set $U = \{u_i'\colon i\in T\}$ satisfies
\begin{enumerate}[label=(\alph*)]
\item $\mathcal{H}[U] = \mathcal{G}[U] \cong K_{t+2}^{3}$,
\item $L(v)[U\setminus\{u_1\}]\cong K_{t+1}$, and
\item $L(u_i)[U] = L_{\mathcal{G}}(u_i)[U]$ for $i\in \{t+1,t+2\}$.
\end{enumerate}

Let $F = \mathcal{H}[U\cup \{v,u_{t+1},u_{t+2}\}]$ and $U' = \left(U\cup \{u_{t+1}\}\right)\setminus \{u_{t+1}'\}$.
Since $v(F) = t+5 \le 4(t+4)^2$,
by assumption, there is a homomorphism $\phi\colon V(F) \to V(\Gamma_{t+2})$ from $F$ to $\Gamma_{t+2}$.
Since $\mathcal{H}[U] \cong \mathcal{H}[U'] \cong K_{t+2}^{3}$,
we have $\Gamma_{t+2}[\phi(U)] \cong \Gamma_{t+2}[\phi(U')] \cong K_{t+2}^3$.
Since $|\phi(U) \cap \phi(U')| = |U\cap U'| = t+1$,
it follows from Lemma~\ref{LEMMA:two-K_t+2} that $\phi(U) = \phi(U')$ and $\phi(u_{t+1}) = \phi(u_{t+1}')$.

Let $U_1 = U\cup\{u_{t+2}\}$.
Since $F[U_1]\cong \Gamma_{t+2}\setminus\{t+3\}$, it follows from Lemma~\ref{LEMMA:gamma-t-partial-embedding}
that $\phi(\{u_i'\colon i\in[t]\}) = [t]$ (and $\phi(u'_{t}) = t$ if $t\ge 3$),
$\phi(\{u_{t+1}',u_{t+2},u_{t+4}'\}) \subset [t+1,t+4]$,
and $\phi(\{u_{t+1}',u_{t+4}'\}) \in \{\{t+1,t+4\},\{t+2,t+3\}\}$.
By symmetry we may assume that $\phi(u_{i}') = i$ for $i\in S$ and $\phi(u_{t+2}) = t+2$.
Hence $\phi(u_{t+1}') = \phi(u_{t+1}) = t+1$.
Since $v$ is adjacent to all vertices in $U_1\setminus\{u_{1}\}$,
we have $\phi(v)\in [t+4]\setminus\phi(U_1\setminus\{u_{1}\}) = \{1,t+3\}$.
If $\phi(v) = t+3$, then $\{v,u_{t+1},u_{t+2}\}\in F$ implies that
$\{t+1,t+2,t+3\} = \{\phi(v),\phi(u_{t+1}),\phi(u_{t+2})\} \in \Gamma_{t+2}$, a contradiction.
Therefore, we may assume that $\phi(v) = 1$.
Then $\left\{\{u_1',u_{t+1}',u_{t+4}'\},\{u_1',u_{t+2},u_{t+4}'\},\{v,u_{t+1},u_{t+2}\}\}\right\}\subset F$ implies
that
\begin{align}
& \left\{\{1,t+1,t+4\},\{1,t+2,t+4\},\{1,t+1,t+2\}\}\right\} \notag\\
=&\left\{\{\phi(u_1'),\phi(u_{t+1}'),\phi(u_{t+4}')\},
        \{\phi(u_1'),\phi(u_{t+2}),\phi(u_{t+4}')\},\{\phi(v),\phi(u_{t+1}),\phi(u_{t+2})\}\}\right\}
\subset\Gamma_{t+2}. \notag
\end{align}
This means that the triangle $\{t+1,t+2,t+4\}$ is contained in the induced subgraph of $L_{\Gamma_{t+2}}(1)$ on $\{t+1, t+2, t+3, t+4\}$,
contradicts the fact that the induced subgraph of $L_{\Gamma_{t+2}}(1)$ on $\{t+1, t+2, t+3, t+4\}$ is a copy of $C_4$.
\end{proof

\medskip

Recall that $S = [t]\cup \{t+1, t+2\}$

\textbf{Case 2.2}: $I_{small}\cap S = \emptyset$.

Recall that $R$ is a graph on $[t+4]$ in which two vertices $i$ and $j$ are adjacent
iff there exists $e\in L(v)$ such that $e\cap V_{i} \neq \emptyset$ and $e\cap V_{j} \neq \emptyset$.
Recall the observation that $L(v)\subset R[V_1,\ldots,V_{t+4}]$.


\begin{claim}\label{CALIM:case2-Vj+2-nonempty}
If $V_{t+2}\cup V_{t+3}\neq\emptyset$, then
        either $L(v) \subset L_{\mathcal{G}}(t+2)$ or $L(v) \subset L_{\mathcal{G}}(t+3)$.
\end{claim}
\begin{proof}
For every $ij\in R$ let $e_{ij} \in L(v)$ be an edge such that $e_{ij}\cap V_{i}\neq \emptyset$
and $e_{ij}\cap V_{j}\neq \emptyset$.
Let $U_1 = \bigcup_{ij\in R_2}e_{ij}$ be a vertex subset of $V$. Note that $|U_1|\le 2\binom{t+4}{2}$.

By symmetry we may assume that $V_{t+2} \neq \emptyset$.
Fix a vertex $u_{t+2} \in V_{t+2}$ (it is possible that $u_{t+2}\in U_1$ as well).
For every $i\in S$ let $V_i' = V_i\cap N(v) \cap N(u_{t+2}) \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right)$.
Notice that $|V_i'| \ge \frac{n}{2(t+2)} - (2\binom{t+4}{2}+1)12t\epsilon > 4(t+4)^3\epsilon^{1/3}n$.
Applying Lemma~\ref{LEMMA:greedily-embedding-Gi} with $\eta = 4t\epsilon$ and $T = S$
we obtain $u_{i}\in V'_i$ for every $i\in T$ such that the set $U = \{u_i\colon i\in T\}$ satisfies
\begin{enumerate}[label=(\alph*)]
\item $\mathcal{H}[U] = \mathcal{G}[U] \cong K_{t+2}^{3}$,
\item $L(u)[U] = L_{\mathcal{G}}(u)[U]$ for all $u\in U_1$, and
\item $L(u_{t+2})[U] = L_{\mathcal{G}}(u_{t+2})[U]$.
\end{enumerate}

Let $F = \mathcal{H}[U\cup U_1\cup \{v,u_{t+2}\}]$. Since $v(F) \le 2\binom{t+4}{2}+t+5 \le 4(t+4)^3$,
by assumption there is a homomorphism $\phi\colon V(F) \to V(\Gamma_{t+2})$ from $F$ to $\Gamma_{t+2}$.
Since $F[U\cup \{u_{t+2}\}] \cong \Gamma_{t+2}\setminus\{t+3\}$, it follows from Lemma~\ref{LEMMA:gamma-t-partial-embedding}
that $\phi(\{u_i\colon i\in [t]\}) = [t]$ (and $\phi(u_{t}) = t$ if $t\ge 3$),
$\phi(\{u_{t+1},u_{t+2},u_{t+4}\}) \subset [t+1,t+4]$, and
$\phi(\{u_{t+1},u_{t+4}\})\in \{\{t+1,t+4\},\{t+2,t+3\}\}$.
By symmetry, we may assume that $\phi(u_{i}) = i$ for $i\in [t]\cup \{t+1,t+2,t+4\}$.
For every $i\in [t]\cup \{t+1,t+4\}$ and every $u\in U_{1}\cap V_{i}$,
since $L(u)[U] = L(u_i)[U] = L_{\mathcal{G}}(u_i)[U]$
and $u$ is adjacent to all vertices in $U\setminus \{u_i\}$,
we have $\phi(u) = \phi(u_i) = i$.
Finally, since $v$ is adjacent to all vertices in $\{u_i\colon i\in [t]\cup \{t+1,t+3\}\}$,
we have $\phi(v) \in [t+4]\setminus\phi(\{u_i\colon i\in [t]\cup \{t+1,t+3\}\}) = \{t+2,t+3\}$.
If $\phi(v) = t+2$, then $\phi(R) \subset L_{\Gamma_{t+2}}(\phi(v)) = L_{\Gamma_{t+2}}(t+2)$,
which means that $L(v)\subset L_{\mathcal{G}}(t+2)$.
If $\phi(v) = t+3$, then $\phi(R) \subset L_{\Gamma_{t+2}}(\phi(v)) = L_{\Gamma_{t+2}}(t+3)$,
which means that $L(v)\subset L_{\mathcal{G}}(t+3)$.
\end{proof

Now we may assume that $V_{j+2}\cup V_{j+3}=\emptyset$.

\begin{claim}\label{CLAIM:case2-structure-of-R}
We have either $L(v)\subset L_{\mathcal{G}}(t+2)$ or $L(v)\subset L_{\mathcal{G}}(t+3)$.
\end{claim}
\begin{proof}
For each edge $ij \in R$ let $e_{i,j} \in L(v)$ be an edge with one endpoint in $V_i$
and the other endpoint in $V_j$.
Let $U_1 = \bigcup_{ij\in R}e_{i,j}$ be a vertex subset of $V$ and note that $|U_1| \le 2\binom{t+2}{2}$.
Define
\begin{align}
V_i' = V_i \cap N(v) \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right) \quad\text{for } i \in [t]\cup \{t+1,t+4\}. \notag
\end{align}
Notice that $|V_i'| \ge \frac{n}{2(t+2)} - (2\binom{t+4}{2}+1)12t\epsilon > 4(t+4)^3\epsilon^{1/3}n$.
Applying Lemma~\ref{LEMMA:greedily-embedding-Gi} with $\eta = 4t\epsilon$ and $T = [t]\cup \{t+1,t+4\}$
we obtain $u_i \in V_i'$ for $i\in T$ such that the set $U_2 = \{u_i\colon i\in T\}$ satisfies
\begin{enumerate}[label=(\alph*)]
  \item $\mathcal{H}[U_2] =\mathcal{G}[U_2] \cong K_{t+2}^3$, and
  \item $L(u)[U_2\setminus\{u_i\}] =L_{\mathcal{G}}(u)[U_2\setminus\{u_i\}]\cong K_{t+1}$
  for all $u\in U_1\cap V_i$ and $i\in S$.
\end{enumerate}
Let $F = \mathcal{H}[U_1\cup U_2 \cup \{v\}]$.
Suppose that there exists a homomorphism $\phi\colon V(F) \to V(\Gamma_{t+2})$ from $F$ to $\Gamma_{t+2}$.
Since the set $F[U_2] \cong K_{t+2}^3$ we have either $\phi(U_2) = [t]\cup\{t+1,t+4\}$
or $\phi(U_2) = [t]\cup\{t+2,t+3\}$.
By symmetry we may assume that the former case holds and $\phi(u_i) = i$ for $i\in [t]\cup\{t+1,t+4\}$.
Since $L_{\Gamma_{t+2}}(t+2)$ and $L_{\Gamma_{t+2}}(t+3)$ do not contain $K_{t+1}$ as a subgraph,
it follows from $(b)$ that for every $u\in V_i\cap U_1$ and $i\in [t]\cup\{t+1,t+4\}$
we have $\phi(u) \not\in\{t+2,t+3\}$.
Moreover, since $u$ is adjacent to all vertices in $U_2\setminus \{u_i\}$ in $\partial F$,
we have $\phi(u)\not\in \phi(U_2\setminus \{u_i\})$ as well.
Therefore, $\phi(u) = \phi(u_i) = i$.
Finally, since $U_2\cup \{v\}$ is $2$-covered in $F$, we have $\phi(v) \in \{t+2,t+3\}$.
If $\phi(v) = t+2$, then we have $L(v)\subset L_{\mathcal{G}}(t+2)$.
If $\phi(v) = t+3$, then we have $L(v)\subset L_{\mathcal{G}}(t+3)$.
\end{proof
This completes the proof of Theorem~\ref{THM:main-sec1.1}~(d).
\end{proof
\section{Feasible region}\label{SEC:proof-feasible-region}
We prove Theorem~\ref{THM:feasibe-region} in this section.
First we need the following simple corollary of Theorem~\ref{THM:main-sec1.1}~(d).
For convenience, we will keep using $t+2$ instead of $t$ in this section.

Recall that for an $n$-vertex $r$-graph $\mathcal{H}$ the edge density of $\mathcal{H}$ is $\rho(H) = |\mathcal{H}|/\binom{n}{r}$, and the shadow density of $\mathcal{H}$ is $|\partial\mathcal{H}|/\binom{n}{r-1}$.

\begin{corollary}\label{CORO:stability}
For every integer $t\ge 2$
there exist constants $\epsilon_0>0$ and $N_0$ such that the following statement holds for all $\epsilon\le \epsilon_0$
and $n\ge N_0$.
Suppose that $\mathcal{H}$ is an $n$-vertex $\mathcal{F}_{t+2}$-free $3$-graph with
$\rho(\mathcal{H})\ge  \frac{t(t+1)}{2(t+2)^2} - \epsilon$.
Then there exists a set $Z_{\epsilon}\subset V(\mathcal{H})$ of size at most $\epsilon^{1/2}n$
such that $\mathcal{H}\setminus Z_{\epsilon}$ is $\Gamma_{t+2}$-colorable and $\delta(\mathcal{H}) \ge \left( \frac{t(t+1)}{2(t+2)^2} - 3\epsilon^{1/2}\right)n^2$.
\end{corollary}

The idea for proving Corollary~\ref{CORO:stability} is to show that after removing a few vertices with small degree
from $\mathcal{H}$ the remaining $3$-graph has large minimum degree, and hence, by Theorem~\ref{THM:main-sec1.1}~(d),
it is $\Gamma_{t+2}$-colorable.
We refer the reader to the proof of Theorem~4.1 in~\cite{LMR1} for detailed calculations.

\begin{lemma}\label{LEMMA:feasible-region-shadow}
Suppose that $\mathcal{H}$ is an $n$-vertex $\Gamma_{t+2}$-colorable $3$-graph with
$|\mathcal{H}| \ge \left(\frac{t(t+1)}{6(t+2)^2}-\epsilon\right)n^3$.
Then $\left(\frac{t+1}{2(t+2)}-100t^4\epsilon^{1/2}\right)n^2 \le
|\partial\mathcal{H}| \le \left(\frac{t^2+3 t+3}{2 (t+2)^2}+5000t^4\epsilon^{1/2}\right)n^2$.
\end{lemma}
\begin{proof}
Let $V = V_1\cup\cdots\cup V_{t+4}$ be a partition such that $\mathcal{H}\subset \Gamma_{t+2}[V_1,\ldots,V_{t+4}]$.
Let $\mathcal{G} = \Gamma_{t+2}[V_1,\ldots,V_{t+4}]$.
Let $x_i = |V_i|/n$ for $i\in[t+4]$.
By Lemma~\ref{LEMMA:Lagrangian-stability}, there exists some $\alpha\in[0,1]$ such that
\begin{align}\label{equ:weight-stable-2}
x_i =
\begin{cases}
\frac{1}{t+2} \pm 30t\epsilon^{1/2} & \text{if } i\in [t], \\
\frac{\alpha}{t+2} \pm 30t\epsilon^{1/2} & \text{if } i\in \{t+1, t+4\},\\
\frac{1-\alpha}{t+2} \pm 30t\epsilon^{1/2} & \text{if } i\in\{t+2, t+3\}.
\end{cases} \notag
\end{align}
Notice that for every pair $\{u,v\}\in \partial\mathcal{G}$
the number of edges in $\mathcal{G}$ containing $\{u,v\}$ is at least
$\left(\frac{1}{t+2} - 30t\epsilon^{1/2}\right)n \ge \frac{n}{2(t+2)}$.
Therefore, it follows from a simple double counting that
\begin{align}
|\partial\mathcal{G}\setminus \partial\mathcal{H}|
\le \frac{3|\mathcal{G}\setminus\mathcal{H}|}{n/2(t+2)}
\le \frac{3\epsilon n^3}{n/2(t+2)}
\le 6\epsilon (t+2)n^2. \notag
\end{align}
Therefore,
\begin{align}
|\partial\mathcal{H}|/n^2
& \ge |\partial\mathcal{G}|/n^2 -6\epsilon (t+2)
 = p_{\partial\mathcal{G}}(x_1,\ldots,x_{t+4})-6(t+2)\epsilon \notag\\
& \ge \binom{t}{2}\left(\frac{1}{t+2} - 30t\epsilon^{1/2}\right)^2
        + t\left(\frac{1}{t+2} - 30t\epsilon^{1/2}\right)\left(\frac{2}{t+2} - 4\times 30t\epsilon^{1/2}\right) \notag\\
& \quad + \left(\frac{\alpha}{t+2} - 30t\epsilon^{1/2}\right)^2
        +\left(\frac{1-\alpha}{t+2} - 30t\epsilon^{1/2}\right)^2 \notag\\
& \quad   + 4\left(\frac{\alpha}{t+2} - 30t\epsilon^{1/2}\right)\left(\frac{1-\alpha}{t+2} - 30t\epsilon^{1/2}\right)
        -6(t+2)\epsilon \notag\\
& \ge \frac{t+1}{2(t+2)}+\frac{2\alpha(1-\alpha)}{(t+2)^2} -30t(t+3)\epsilon^{1/2}+450 t^2 \left(t^2+7 t+12\right)\epsilon-6(t+2)\epsilon\notag\\
& \ge \frac{t+1}{2(t+2)} - 100t^4\epsilon^{1/2}. \notag
\end{align}
On the other hand, we have
\begin{align}
|\partial\mathcal{H}|/n^2
& \le |\partial\mathcal{G}|/n^2
 = p_{\partial\mathcal{G}}(x_1,\ldots,x_{t+4}) \notag\\
& \le \binom{t}{2}\left(\frac{1}{t+2} +30t\epsilon^{1/2}\right)^2
        + t\left(\frac{1}{t+2} + 30t\epsilon^{1/2}\right)\left(\frac{2}{t+2} + 4\times 30t\epsilon^{1/2}\right) \notag\\
& \quad + \left(\frac{\alpha}{t+2} + 30t\epsilon^{1/2}\right)^2
        +\left(\frac{1-\alpha}{t+2} + 30t\epsilon^{1/2}\right)^2 \notag\\
& \quad   + 4\left(\frac{\alpha}{t+2} + 30t\epsilon^{1/2}\right)\left(\frac{1-\alpha}{t+2} + 30t\epsilon^{1/2}\right) \notag\\
& \le \frac{t+1}{2(t+2)}+\frac{2\alpha(1-\alpha)}{(t+2)^2}+30 t (t+3) \epsilon^{1/2}+450 t^2 \left(t^2+7 t+12\right)\epsilon \notag\\
& \le \frac{t+1}{2(t+2)}+\frac{1}{2(t+2)^2} + 5000t^4\epsilon^{1/2}
 = \frac{t^2+3 t+3}{2 (t+2)^2}+ 5000t^4\epsilon^{1/2}. \notag
\end{align}
\end{proof}

Now we are ready to prove Theorem~\ref{THM:feasibe-region}.

\begin{proof}[Proof of Theorem~\ref{THM:feasibe-region}]
First, we prove that $\mathrm{proj}\Omega(\mathcal{F}_{t+2}) = \left[0,\frac{t+3}{t+4}\right]$.
By a result in~\cite{LM1}, this is equivalent to show that $\frac{t+3}{t+4} \in \mathrm{proj}\Omega(\mathcal{F}_{t+2})$ and $\mathrm{proj}\Omega(\mathcal{F}_{t+2}) \subset \left[0,\frac{t+3}{t+4}\right]$.
Let $\mathcal{H}$ be an $n$-vertex $\mathcal{F}_{t+2}$-free $3$-graph.
By Lemma~\ref{LEMMA:F-free-shadow-Km-free}, the graph $\partial\mathcal{H}$ is $K_{t+5}$-free.
Thus, by Tur\'{a}n's theorem, we have $\rho(\partial\mathcal{H}) \le \frac{t+3}{t+4}$.
Therefore, we have $\mathrm{proj}\Omega(\mathcal{F}_{t+2})\subset \left[0,\frac{t+3}{t+4}\right]$.
On the other hand, since the balanced blowup of $\Gamma_{t+2}$ on $n$ vertices has edge density $\frac{t+3}{t+4}$ as $n\to \infty$,
we have $\frac{t+3}{t+4} \in \mathrm{proj}\Omega(\mathcal{F}_{t+2})$.
Therefore, $\mathrm{proj}\Omega(\mathcal{F}_{t+2}) = \left[0,\frac{t+3}{t+4}\right]$.

Let $I_t = \left[\frac{t+1}{t+4},\frac{t^2+3 t+3}{(t+2)^2}\right]$.
Next, we show that $I_t \times \left\{\frac{t (t+1)}{(t+2)^2}\right\}\subset \Omega(\mathcal{F}_{t+2})$.
This is done by constructing for every $x\in I_t$ a sequence of $\mathcal{F}_{t+2}$-free $3$-graphs whose shadow densities approach $x$, and whose edge densities approach $\frac{t (t+1)}{(t+2)^2}$.
Let $\alpha \in [0,1/2]$ be a real number.
Let $x_1 = \cdots= x_t = \frac{1}{t+2}$, $x_{t+1} = x_{t+4} = \frac{\alpha}{t+2}$, and
$x_{t+2} = x_{t+3} = \frac{1-\alpha}{t+2}$.
For every $n\in \mathbb{N}$ let
$\mathcal{H}_n(\alpha) = \Gamma_{t+2}[V_1,\ldots,V_{t+4}]$ be a blowup of $\Gamma_{t+2}$
such that $|V_i| = \lfloor x_i n \rfloor$ for $i\in [t+4]$.
It is easy to see that
\begin{align}
\lim_{n\to\infty}\rho\left(\partial\mathcal{H}_n(\alpha)\right)
= 2 \cdot p_{K_{t+4}}(x_1,\ldots,x_{t+4})
= \frac{t^2+3 t+2+4\alpha(1-\alpha)}{(t+2)^2}, \notag
\end{align}
and
\begin{align}
\lim_{n\to\infty}\rho\left(\mathcal{H}_n(\alpha)\right)
= 6\cdot p_{\Gamma_{t+2}}(x_1,\ldots,x_{t+4})
= \frac{t (t+1)}{(t+2)^2}. \notag
\end{align}
Letting $\alpha$ vary from $0$ to $1/2$,
the function $\frac{t^2+3 t+2+4\alpha(1-\alpha)}{2 (t+2)^2}$ grows from $\frac{t+1}{t+4}$ to $\frac{t^2+3 t+3}{(t+2)^2}$. Therefore,
$I_t \times \left\{\frac{t (t+1)}{6 (t+2)^2}\right\}\subset \Omega(\mathcal{F}_{t+2})$.

Finally, we show that the feasible region function $g(\mathcal{F}_{t+2})$ attains its maximum only on the interval $I_t$.
Suppose that $\left(\mathcal{H}_n\right)_{n=1}^{\infty}$ is a sequence of $\mathcal{F}_{t+2}$-free $3$-graphs
with $\lim_{n\to \infty}\rho(\partial\mathcal{H}_n) = x$ and $\lim_{n\to \infty}\rho(\mathcal{H}_n) = \frac{t(t+1)}{(t+2)^2}$.
To keep our notations simple, let us assume that $v(\mathcal{H}_n) = n$ for all $n\ge 1$.
Let $\epsilon>0$ be a sufficiently small constant.
Then there exists $N_0$ such that $\rho(\partial\mathcal{H}_n) = x\pm \epsilon$ and
$\rho(\mathcal{H}_n) \ge \frac{t(t+1)}{(t+2)^2} -\epsilon$ for all $n\ge N_0$.
It follows from Corollary~\ref{CORO:stability} that there exists a set $Z_{n}\subset V(\mathcal{H}_n)$
of size $\epsilon^{1/2} n$
such that the $3$-graph $\mathcal{H}'_{n}$ obtained from $\mathcal{H}_n$ by removing all edges that have
nonempty intersection with $Z_n$ is $\Gamma_{t+2}$-colorable with minimum degree at least $\delta(\mathcal{H}) \ge \left( \frac{t(t+1)}{2(t+2)^2} - 3\epsilon^{1/2}\right)n^2$.
Fixing a $\Gamma_{t+2}$-coloring $V(\mathcal{H}_n') = V_{n,1} \cup \cdots \cup V_{n,t+4}$ of $\mathcal{H}_n'$
and let $\mathcal{G}_n = \Gamma_{t+2}[V_{n,1} \cup \cdots \cup V_{n,t+4}]$ be the blowup of $\Gamma_{t+2}$.

Since $\rho(\mathcal{H}_n') \ge \rho(\mathcal{H}_n)-|Z_{\epsilon}|n^2/\binom{n}{3}\ge  \rho(\mathcal{H}_n)-6\epsilon^{1/2} \ge \frac{t(t+1)}{(t+2)^2}-7\epsilon^{1/2}$,
it follows from Lemma~\ref{LEMMA:feasible-region-shadow} that
$$\frac{t+1}{t+2}-2800t^4\epsilon^{1/4} \le
\rho(\partial\mathcal{H}_n') \le \rho(\partial\mathcal{G}_n) \le \frac{t^2+3 t+3}{(t+2)^2}+140000t^4\epsilon^{1/4}.$$
By Claim~\ref{CLAIM:link-empty-inside},
for every $v\in Z_n$ we have $L_{\mathcal{H}_n}(v)\cap V_{n,i} = \emptyset$ \footnote{Note that in the proof of Claim~\ref{CLAIM:link-empty-inside}
we do not require $d_{\mathcal{H}_n}(v)$ to be large.} for all $i\in [t+4]$.
Therefore, $\rho(\partial\mathcal{H}_n) \le \rho(\partial\mathcal{G}_n) + |Z_n|n/\binom{n}{2} \le \rho(\partial\mathcal{G}_n) +3\epsilon^{1/2}$.
So,
\begin{align}
\frac{t+1}{t+2}-2800t^4\epsilon^{1/4}
\le \rho(\partial\mathcal{H}'_n)
\le \rho(\partial\mathcal{H}_n)
\le \rho(\partial\mathcal{G}_n) +3\epsilon^{1/2}
\le \frac{t^2+3 t+3}{(t+2)^2}+140001t^4\epsilon^{1/4}.\notag
\end{align}
Consequently,
\begin{align}
\frac{t+1}{t+2}-2800t^4\epsilon^{1/4}-\epsilon
\le x
\le \frac{t^2+3 t+3}{(t+2)^2}+140001t^4\epsilon^{1/4} +\epsilon.\notag
\end{align}
Letting $\epsilon \to 0$, we have
\begin{align}
\frac{t+1}{t+2}
\le x
\le \frac{t^2+3 t+3}{(t+2)^2}.\notag
\end{align}
This completes the proof of Theorem~\ref{THM:feasibe-region}.
\end{proof}
\section{Concluding remarks}\label{SEC:Remarks}
$\bullet$  We defined the crossed blowup and used it to construct finite family of triple systems with infinitely many extremal constructions.
One could extended this operation in the following way.

Instead of replacing a pair of vertices (i.e. $\{v_1,v_2\}$) by four vertices (i.e. $\{v_1,v_1',v_2,v_2'\}$),
one could replace it by $2^k$ vertices for $k\ge 2$.

Let $Q_{k}$ denote the vertex set of the $k$-dimensional hypercube (each vertex is represented by a length-$k$ binary string).
For $i\in [k]$ let $Q_{k}(i,0)\subset Q_{k}$ and $Q_{k}(i,1)\subset Q_{k}$ denote the collection of vertices whose $i$-th coordinate is
$0$ and $1$, respectively.



\begin{definition}[$k$-crossed blowup]\label{DFNp:k-crosse-blowup}
Let $k\ge 2$ be an integer.
Let $\mathcal{G}$ be a $3$-graph  and $\{v_1,v_2\}\subset V(\mathcal{H})$
be a pair with $d_{\mathcal{G}}(v_1,v_2) = d \ge k$.
Assume that $N_{\mathcal{G}}(v_1,v_2)=\{u_1,  \ldots, u_d\}$.
The $k$-crossed blowup $\mathcal{G}\boxplus^{k}\{v_1,v_2\}$ of $\mathcal{G}$ on $\{v_1,v_2\}$ is obtained in the following way.
\begin{enumerate}[label=(\alph*)]
\item Remove all edges in $\mathcal{G}$ that contain $\{v_1,v_2\}$,
\item replace $\{v_1,v_2\}$ by $2^k$ new vertices $Q_{k}$ so that every vertex in $Q(1,0)$ is a clone of $v_1$ and every vertex in
        $Q(1,1)$ is a colon of $v_2$,
\item for $i\in [k-1]$ we add a set $\mathcal{E}_i$ of triples containing $u_i$ so that $L_{\mathcal{E}_i}(u_i)$ is the
        complete bipartite graph $B[Q(i,0),Q(i,1)]$ with two parts $Q(i,0),Q(i,1)$,
\item for $i \in [k,d]$ we add  a set $\mathcal{E}_i$ of triples containing $u_i$ so that $L_{\mathcal{E}_i}(u_i)$ is the
        complete bipartite graph $B[Q(k,0),Q(k,1)]$ with two parts $Q(k,0),Q(k,1)$.
\end{enumerate}
\end{definition}

\begin{figure}[htbp]
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw  [fill={rgb, 255:red, 255; green, 255; blue, 255 }  ,fill opacity=1 ][dash pattern={on 4.5pt off 4.5pt}] (431.08,189.69) -- (464.73,156.05) -- (555.75,156.05) -- (555.75,234.56) -- (522.1,268.21) -- (431.08,268.21) -- cycle ; \draw  [dash pattern={on 4.5pt off 4.5pt}] (555.75,156.05) -- (522.1,189.69) -- (431.08,189.69) ; \draw  [dash pattern={on 4.5pt off 4.5pt}] (522.1,189.69) -- (522.1,268.21) ;
\draw [color={rgb, 255:red, 184; green, 233; blue, 134 }  ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}]  (466.83,234.56) -- (601.92,234.96) ;
\draw [shift={(603.92,234.97)}, rotate = 180.17] [color={rgb, 255:red, 184; green, 233; blue, 134 }  ,draw opacity=1 ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}]  (466.72,234.56) -- (406.12,292.51) ;
\draw [shift={(404.68,293.89)}, rotate = 316.28] [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 }  ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}]  (466.72,234.56) -- (465.9,106.67) ;
\draw [shift={(465.89,104.67)}, rotate = 89.63] [color={rgb, 255:red, 74; green, 144; blue, 226 }  ,draw opacity=1 ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw  [color={rgb, 255:red, 248; green, 231; blue, 28 }  ,draw opacity=1 ][fill={rgb, 255:red, 248; green, 231; blue, 28 }  ,fill opacity=1 ] (305.99,191.43) .. controls (305.99,189.41) and (307.75,187.77) .. (309.91,187.77) .. controls (312.07,187.77) and (313.82,189.41) .. (313.82,191.43) .. controls (313.82,193.45) and (312.07,195.08) .. (309.91,195.08) .. controls (307.75,195.08) and (305.99,193.45) .. (305.99,191.43) -- cycle ;
\draw  [color={rgb, 255:red, 184; green, 233; blue, 134 }  ,draw opacity=1 ][fill={rgb, 255:red, 184; green, 233; blue, 134 }  ,fill opacity=1 ] (306,223.91) .. controls (306,221.89) and (307.75,220.25) .. (309.91,220.25) .. controls (312.07,220.25) and (313.82,221.89) .. (313.82,223.91) .. controls (313.82,225.93) and (312.07,227.57) .. (309.91,227.57) .. controls (307.75,227.57) and (306,225.93) .. (306,223.91) -- cycle ;
\draw  [color={rgb, 255:red, 74; green, 144; blue, 226 }  ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 }  ,fill opacity=1 ] (306,256.39) .. controls (306,254.37) and (307.75,252.74) .. (309.91,252.74) .. controls (312.07,252.74) and (313.82,254.37) .. (313.82,256.39) .. controls (313.82,258.41) and (312.07,260.05) .. (309.91,260.05) .. controls (307.75,260.05) and (306,258.41) .. (306,256.39) -- cycle ;
\draw (320.92,182.12) node [anchor=north west][inner sep=0.75pt]   [align=left] {1};
\draw (320.05,216.23) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
\draw (320.05,247.9) node [anchor=north west][inner sep=0.75pt]   [align=left] {3};
\draw [fill=uuuuuu] (431.08,189.69) circle (1.5pt);
\draw [fill=uuuuuu] (464.73,156.05) circle (1.5pt);
\draw [fill=uuuuuu] (555.75,156.05) circle (1.5pt);
\draw [fill=uuuuuu] (555.75,234.56) circle (1.5pt);
\draw [fill=uuuuuu] (522.1,268.21) circle (1.5pt);
\draw [fill=uuuuuu] (431.08,268.21) circle (1.5pt);
\draw [fill=uuuuuu] (466.83,234.56)  circle (1.5pt);
\draw [fill=uuuuuu] (522.1,189.69) circle (1.5pt);
\draw (461.8,234.47) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {000};
\draw (556.04,235.77) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {010};
\draw (425.32,270.2) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {100};
\draw (520.11,269.42) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {110};
\draw (465.28,142.7) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {001};
\draw (556.57,142.7) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {011};
\draw (431.37,190.91) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {101};
\draw (520.18,190.91) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {111};
\end{tikzpicture}
\caption{$\{145,245,345\}\boxplus^3\{4,5\}$, in which $L(1)$ is the complete bipartite graph with
         parts $\{000,001,010,011\}$ and $\{100,101,110,111\}$ (these two squares are perpendicular to the yellow axis).
         Similarly to $L(2)$ and $L(3)$.}
\end{figure}

Notice that the crossed blowup we defined in Section~\ref{SEC:Introduction} is $2$-crossed blowup.
Using a similar argument, one could extend
Proposition~\ref{PROP:nonminimal-corss-blowup} to $k$-crossed blowups for all $k\ge 2$.

There is also a natural extension to $r$-graphs for every $r\ge 4$. We omit the definition here.

\medskip

$\bullet$ There are two ways to extend Theorem~\ref{THM:main-sec1.1} to $r$-graphs for every $r\ge 4$.
One is to use $\Gamma_{t+2}$ to construct an $r$-graph in the following way:
take $r-3$ new vertices $\{u_1,\ldots, u_{r-3}\}$ and let
\begin{align}
\Gamma_{t+2}^r = \left\{\{u_1,\ldots, u_{r-3}\}\cup E\colon E\in \Gamma_{t+2}\right\}. \notag
\end{align}
Following the argument as in~\cite{LMR3} one could extend Theorem~\ref{THM:main-sec1.1} to $r$-graphs.

Another way is to consider the crossed blowup of the $r$-uniform complete graph $K_{t+2}^{r}$.
Following the argument as in the present paper one could extend Theorem~\ref{THM:main-sec1.1} to $r$-graphs as well.

\medskip

$\bullet$ For $t=1$ one could use the fact that the set $Z(\Gamma_1)$ is a two-dimensional simplex
\begin{align}
\left\{\alpha \vec{x}_1 +\beta \vec{x}_2 +(1-\alpha-\beta) \vec{x}_3
        \colon \alpha\ge 0, \beta\ge 0, \alpha + \beta\le 1\right\}, \notag
\end{align}
where $\vec{x}_1 = \left(\frac{1}{3},0,\frac{1}{3},0,0,\frac{1}{3}\right)$,
$\vec{x}_2 = \left(0,\frac{1}{3},\frac{1}{3},0,0,\frac{1}{3}\right)$, and
$\vec{x}_3 = \left(0,\frac{1}{3},0,\frac{1}{3},\frac{1}{3},0,\right)$,
to improve Theorem~\ref{THM:main-sec1.1}~(b) to show that
the number of maximum $\mathcal{F}_1$-free $3$-graphs on $n$ vertices is $\Theta(n^2)$.
\bibliographystyle{abbrv}
