On the sum of sizes of overlapping families

TL;DR

This paper investigates the maximum total size of families of k-subsets of an n-set under the restriction that no s+1 families contain pairwise disjoint sets, relating to the Erdős Matching Conjecture and providing new bounds and conjectures.

Contribution

It offers new upper bounds, proposes a general conjecture, and solves it for the case when n is at least 4k^2s, advancing understanding of extremal set configurations.

Findings

Provided upper bounds for the sum of family sizes.

Formulated a general conjecture on the maximum sum.

Solved the conjecture for n ≥ 4k^2s.

Abstract

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ 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 $|\mathcal{A}_1|+\ldots+|\mathcal{A}_m|$. Note that the subcase $m=s+1$, $\mathcal{A}_1=\ldots=\mathcal{A}_m$ is the Erd\H{o}s 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 $n\geq 4k^2s$.