Combinatorial Theory

We say that the families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,\dots ,F_{s+1}$, where $F_i\in {\mathcal{F}}_i$ for each $i$. The rainbow version of the Erdős Matching Conjecture due to Aharoni and Howard and independently to Huang, Loh and Sudakov states that $\underset{i}{min}|{\mathcal{F}}_i|\le max\{(\genfrac{}{}{0ex}{}{n}{k})-(\genfrac{}{}{0ex}{}{n-s}{k}),(\genfrac{}{}{0ex}{}{(s+1)k-1}{k})\}$ 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 Kupavskii.

Mathematics Subject Classifications: 05D05

Keywords: Extremal set theory, Erdos matching conjecture, rainbow version