A Sharp Bound for the Number of Sets that Pairwise Intersect at k Positive Values

In this paper we prove that if ${\user1{L}}$ is a set of k positive integers and {A ₁, ..., A _m } is a family of subsets of an n-element set satisfying ${\left| {A_{i} \cap A_{j} } \right|} \in {\user1{L}}$, for all 1 ≤ i < j ≤ m, then $m \leqslant {\sum\nolimits_{i = 0}^k {{\left( {^{{n - 1}}_{i} } \right)}} }$. The case k = 1 was proven 50 years ago by Majumdar.