I am trying to determine the maximum possible size of a collection of $k$-element subsets of {$1, 2, \cdots n$} set whose pairwise intersections are at most 2.
It's clear that when $k = 3$, its just the number of distinct three element subsets of {$1, 2, \cdots n$} = $\binom{n}{3}$. I've also been counting them out for smaller cases and don't see the pattern.
Edit: I've searched through all the suggested questions that arose when I was asking my question and found nothing that helps me find bounds or a precise solution.