More on the Erd\H os--Kleitman problem on matchings in set families

TL;DR

This paper investigates the maximum size of set families avoiding s pairwise disjoint subsets, building on classical and recent results, and proves an approximate version of a conjecture related to the Erdős Matching Conjecture.

Contribution

It proves an approximate version of Frankl and Kupavskii's conjecture for large s, extending understanding of extremal set families related to the Erdős Matching Conjecture.

Findings

Established an approximate version of the conjecture for s ≥ s₀(m).

Connected extremal examples to the Erdős Matching Conjecture.

Extended classical results to new parameter ranges.

Abstract

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erd\H{o}s, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erd\H{o}s Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.