The Erdős Matching Conjecture and concentration inequalities

Abstract

More than 50 years ago, Erdős asked the following question: what is the maximum size of family 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 |F|≤(nk)−(n−sk), provided n≥53sk−23s and s is sufficiently large. The bound on |F| is sharp since the family of all k-sets that intersect some fixed s-element set has such size and has no s+1 pairwise disjoint sets. We derive several corollaries concerning Dirac thresholds and deviations of sums of random variables. We also obtain several related results.