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Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.

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On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty ...

Lucia's answer given in the linked question from the comment gives an upper bound (i.e., no pair should appear twice as a subset of $A_i$). But of course, the real question starts from here: When can …
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