Two problems on matchings in set families - in the footsteps of Erd\H{o}s and Kleitman

The families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_s\subset 2^{[n]}$ are called $q$-dependent if there are no pairwise disjoint $F_1\in \mathcal F_1,\ldots, F_s\in\mathcal F_s$satisfying$|F_1\cup\ldots\cup F_s|\le q.$Wedetermine$\max |\mathcal F_1|+\ldots +|\mathcal F_s| $forallvalues$n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The well-known Erd\H os Matching Conjecture suggests the largest size of a family $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ with no $s$ pairwise disjoint sets. After more than 50 years its full solution is still not in sight. In the present paper, we provide a Hilton-Milner-type stability theorem for the Erd\H{o}s Matching Conjecture in a relatively wide range, in particular, for $n\ge (2+o(1))sk$ with $o(1)$ depending on $s$ only. This is a considerable improvement of a classical result due to Bollob'as, Daykin and Erd\H{o}s. We apply our results to advance in the following anti-Ramsey-type problem, proposed by "Ozkahya and Young. Let $ar(n,k,s)$ be the minimum number $x$ of colors such that in any coloring of the $k$-element subsets of $[n]$ with $x$ (non-empty) colors there is a $\mathit{\text{rainbow}}\mathit{\text{matching}}$ of size $s$, that is, $s$ sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine $ar(n,k,s)$ for all $k\ge 3$ and $n\ge sk+(s-1)(k-1)\mathrm{.}$ Some other consequences of our results are presented as well.