On the sum of sizes of overlapping families

Abstract

Let A1,…,Am be families of k-subsets of an n-set. Suppose that one cannot choose pairwise disjoint edges from s+1 distinct families. Subject to this condition we investigate the maximum of |A1|+…+|Am|. Note that the subcase m=s+1, A1=…=Am is the Erdős Matching Conjecture, one of the most important open problems in extremal set theory. We provide some upper bounds, a general conjecture and its solution for the range n≥4k2s.