Determine the maximal size of a set of positive integers with the following properties:

$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

$2.$ No digit occurs more than once in the same integer.

$3.$ The digits in each integer are in increasing order.

$4.$ Any two integers have at least one digit in common (possibly at different positions).

$5.$ There is no digit which appears in all the integers.

Let's think step by step and output the final answer within \boxed{}.
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