Erd\H{o}s Matching Conjecture for almost perfect matchings

TL;DR

This paper advances the understanding of Erdős's matching conjecture by providing new bounds for the maximum size of certain families of subsets, specifically for large s relative to k, and narrow n ranges.

Contribution

It improves upon recent results by Frankl and resolves the conjecture for large s and specific n ranges, extending the known cases.

Findings

Established new bounds for the maximum size of families without large matchings.

Resolved Erdős's matching conjecture for s > 101k^3 in specified n ranges.

Extended the range of parameters where the conjecture holds.

Abstract

In 1965 Erd\H{o}s asked, what is the largest size of a family of $k$-element subsets of an $n$-element set that does not have a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^{3}$ and $(s+1)k\le n<(s+1)(k+\frac{1}{100k})$.