Two problems on matchings in set families – In the footsteps of Erdős and Kleitman

Abstract

The families F1,…,Fs⊂2[n] are called q-dependent if there are no pairwise disjoint F1∈F1,…,Fs∈Fs satisfying |F1∪…∪Fs|≤q. We determine max⁡|F1|+…+|Fs| for all values n≥q,s≥2. The result provides a far-reaching generalization of an important classical result of Kleitman.The well-known Erdős Matching Conjecture suggests the largest size of a family F⊂([n]k) 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 Erdős Matching Conjecture in a relatively wide range, in particular, for n≥(2+o(1))sk with o(1) depending on s only. This is a considerable improvement of a classical result due to Bollobás, Daykin and Erdős.We apply our results to advance in the following anti-Ramsey-type problem, proposed by Özkahya 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 rainbow 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≥3 and n≥sk+(s−1)(k−1). Some other consequences of our results are presented as well.