Rainbow version of the Erd\H os Matching Conjecture via Concentration

We say that the families $\mathcal F_1,\ldots, \mathcal F_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,\ldots, F_{s+1}$, where $F_i\in \mathcal F_i$ for each $i$. The rainbow version of the Erd\H os Matching Conjecture due to Aharoni and Howard and independently to Huang, Loh and Sudakov states that $\min_{i} |\mathcal F_i|\le \max\big\{{n\choose k}-{n-s\choose k}, {(s+1)k-1\choose k}\big\}$ for $n\ge (s+1)k$. In this paper, we prove this conjecture for $n>3e(s+1)k$ and $s>10^7$. One of the main tools in the proof is a concentration inequality due to Frankl and the author.