A set intersection theorem and an application to a bipartite Ramsey theorem (Q5957764)

The existence of certain subfamilies of families of finite sets is proven. In particular, it is shown that given any integer \(k \geq 2\) there exists an integer \(n\) such that any family of \(n\) \(n\)-sets in \([2n-1]\) contains a subfamily of \(k\) of these \(n\) sets with at least \(k\) elements in common. This result can be used for an alternative proof of a Ramsey-type theorem for bipartite graphs.