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

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