  More than 50 years ago, Erd\H os asked the following question: what is the
maximum size of a family $\mathcal F$ 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
$|\mathcal F|\le {n\choose k}-{n-s\choose k}$, provided $n\ge \frac 53sk -\frac
23 s$ and $s$ is sufficiently large. We derive several corollaries concerning
Dirac thresholds and deviations of sums of random variables. We also obtain
several related results.

Combinatorics