Families without $s$-matchings: the other end

TL;DR

This paper characterizes the maximum size of set families without s pairwise disjoint subsets, extending Erdős Matching Conjecture results to a non-uniform setting for specific parameters.

Contribution

It provides a precise determination of the largest family without s disjoint sets for certain n, generalizing known extremal combinatorics results.

Findings

Largest family size without s disjoint sets determined for n=ms+c.

Extends Erdős Matching Conjecture to non-uniform families.

Results apply when s ≥ s_0(m, c).

Abstract

In this paper, we determine the largest family $\mathcal F \subset 2^{[n]}$ without $s$ pairwise disjoint sets, provided $n=ms+c$ for positive integers $m,c$, and $s \geq s_0(m, c)$. This result can be seen as a non-uniform analogue of the results on the Erd\H os Matching Conjecture in the regime when the clique is extremal.