English

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

Combinatorics 2022-12-20 v2 Discrete Mathematics

Abstract

In 1965 Erd\H{o}s asked, what is the largest size of a family of kk-element subsets of an nn-element set that does not have a matching of size s+1s+1? In this note, we improve upon a recent result of Frankl and resolve this problem for s>101k3s>101k^{3} and (s+1)kn<(s+1)(k+1100k)(s+1)k\le n<(s+1)(k+\frac{1}{100k}).

Keywords

Cite

@article{arxiv.2206.01526,
  title  = {Erd\H{o}s Matching Conjecture for almost perfect matchings},
  author = {Dmitriy Kolupaev and Andrey Kupavskii},
  journal= {arXiv preprint arXiv:2206.01526},
  year   = {2022}
}