The Erd\H{o}s Matching Conjecture and concentration inequalities

TL;DR

This paper advances the understanding of Erdős's matching conjecture by establishing new bounds on the maximum size of certain families of sets, and explores related probabilistic and combinatorial implications.

Contribution

It improves the best known bounds for Erdős's matching conjecture and derives new corollaries related to Dirac thresholds and sum deviations of random variables.

Findings

Proved an improved upper bound on the size of set families avoiding s+1 disjoint sets.

Derived corollaries related to Dirac thresholds in graph theory.

Analyzed deviations of sums of random variables in the context of the conjecture.

Abstract

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint sets? This question attracted a lot of attention recently, in particular, due to its connection to various combinatorial, probabilistic and theoretical computer science problems. Improving the previous best bound due to the first author, we prove that $|\mathcal F|\le {n\choose k}-{n-s\choose k}$, provided $n\ge \frac 53sk -\frac 23 s$ and $s$ is sufficiently large. We derive several corollaries concerning Dirac thresholds and deviations of sums of random variables. We also obtain several related results.