Small Intersections

Let \( S = \{ 1, 2, 3, \ldots 11 \} \) and \( T_1, T_2 \ldots, T_N \) be distinct subsets of \(S\) such that \( | T_i \cap T_j | \leq 2\) for all values \( i\neq j \). What is the maximum possible value of \(N\)?

Details and assumptions

If you are unfamiliar with the set notation symbol \( \cap \), you may refer to Venn Diagram for an explanation of the intersection symbol.

The empty set is a subset of every set.

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