A short proof for the generalized Katona-Kleitman theorem (Q2732545)

The following result generalizing a theorem independently proved by Katona and Kleitman is established in this paper. Let \(S\) be a set of cardinality \(n\) and let \(S_1, S_2, \ldots, S_k\) be a partition of \(S\). If \(F\) is a collection of subsets of \(S\) without elements \(A\) and \(B\) satisfying \(A \cap S_i = B \cap S_i\) for some \(i\) and \(A \cap S_j \subseteq B \cap S_j\) for all \(j \neq i\), then \(|F|\leq {n \choose {\lfloor {n \over 2} \rfloor}}\).