A note on the Erd\H{o}s Matching Conjecture

TL;DR

This paper investigates the Erdős Matching Conjecture for fixed s ≥ 3, large k, and n near sk, establishing stability results and bounds on the maximum size of families without s disjoint sets.

Contribution

It proves stability of the extremal family for n=sk and provides bounds on the maximum family size for n=sk+1 when s is fixed and k tends to infinity.

Findings

Stability of extremal construction at n=sk.

Bound on maximum family size at n=sk+1.

Asymptotic ratio less than (s-1)/s minus a positive constant.

Abstract

The Erd\H os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family $\mathcal{F}\subseteq \binom{[n]}{k}$ that does not contain $s$ pairwise disjoint sets is $\max\{|\mathcal{A}_{k,s}|,|\mathcal{B}_{n,k,s}|\}$, where $\mathcal{A}_{k,s}=\binom{[sk-1]}{k}$ and $\mathcal{B}_{n,k,s}=\{B\in \binom{[n]}{k}:B\cap [s-1]\neq \emptyset\}$. The case $s=2$ is simply the Erd\H{o}s-Ko-Rado theorem on intersecting families and is well understood. The case $n=sk$ was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if $k,s$ are fixed and $n$ is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for $n\in [sk,sk+c_{s,k}]$ if $s$ is large enough compared to $k$. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most $s$ in a similar range of parameters. In this short note, we are concerned with the case $s\ge 3$ fixed, $k$ tending to infinity and $n\in\{sk,sk+1\}$. For $n=sk$, we show the stability of the unique extremal construction of size $\binom{sk-1}{k}=\frac{s-1}{s}\binom{sk}{k}$ with respect to minimal degree. As a consequence we derive $\lim\limits_{k\rightarrow \infty}\frac{f(sk+1,k,s)}{\binom{sk+1}{k}}<\frac{s-1}{s}-\varepsilon_s$ for some positive constant $\varepsilon_s$ which depends only on $s$.