Erdős Matching Conjecture for almost perfect matchings

Abstract

In 1965 Erdős 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 $(s+1)k\le n<(s+1)(k+\frac{1}{100k})$.