On the sum of sizes of overlapping families

Abstract

Let ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_m$ 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 $|{\mathcal{A}}_1|+\dots +|{\mathcal{A}}_m|$. Note that the subcase $m=s+1$, ${\mathcal{A}}_1=\dots ={\mathcal{A}}_m$ 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 $n\ge 4k^{2}s$.

10 pages