Towards the Erd\H{o}s--Kleitman Problem: from Erd\H{o}s matching conjecture perspective

TL;DR

This paper investigates the maximum size of families of subsets with no s pairwise disjoint members, extending the Erdős matching problem, and characterizes extremal families for large s in certain parameter ranges.

Contribution

It proves the structure of extremal families for the Erdős–Kleitman problem when parameters are large, generalizing previous results and determining asymptotic ranges for specific cases.

Findings

Extremal families are of a specific form for large s and certain c.

For m=3, the paper determines the asymptotic range of parameters where a family is extremal.

The results extend the understanding of the Erdős–Kleitman problem for large parameters.

Abstract

For integers $n\ge s\ge2$, let $e(n,s)$ denote the maximum size of a family $\F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is the non-uniform analogue of the Erd\H{o}s matching problem. Fix $m\ge3$ and write $n=sm+c$, $\ell=s-c$. We prove that for every fixed $m\ge3$, there exists constants $\beta_m$ and $\delta_m$ such that for sufficiently large $s$, the extremal families for $e(sm+c,s)$ are $P'(m,s,\ell;L')\coloneqq \binom{L'}m\cup\binom{[sm+c]}{\ge m+1}$ for some $L'$ with $|L'|=m\ell-1$ when $\beta_m s^{(m-1)/m}\le c\le \delta_m s$. For $m=3$, we determine the asymptotic range of $\ell$ for which $P'(3,s,\ell;L')$ is extremal.