Number of subsets of the same size and pairwise intersecting in at most one point
On page number 190 (Section 13.6) of the book "Extremal Combinatorics, Second Edition" by Jukna, the following result, attributed to Frankl and Wilson, is proved:
If $L\subset \mathbb{N}$ is a finite set of integers and if $F$ is a family of subsets of an $n$ element set such that for all $A, B \in F$, $A \neq B$, $|A\cap B| \in L$, then $|F|\leq \sum_{i = 0}^{|L|}\binom{n}{i}$.