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- W2497283602 abstract "The families $mathcal F_1,ldots, mathcal F_ssubset 2^{[n]}$ are called textit{$q$-dependent} if there are no pairwise disjoint $F_iin mathcal F_i, i=1,ldots, s,$ satisfying $|F_1cupldotscup F_s|le q.$ We determine $max |mathcal F_1|+ldots +|mathcal F_s| $ for textit{all} values $nge q,sge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The uniform case $mathcal F_1 = ldots = mathcal F_ssubset {[n]choose k}$ of this problem is the so-called ErdH os Matching Conjecture. 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 it in a relatively wide range, in particular, for $nge (2+o(1))sk$ with $o(1)$ depending on $s$. This is a considerable improvement of a result due to Bollob'as, Daykin and ErdH os. We apply our results to advance in an anti-Ramsey-type problem, proposed by Ozkahya and Young. They asked for the minimum number $ar(n,k,s)$ of colors in the coloring of the $k$-element subsets of $[n]$ that do not contain a textit{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 $kge 3$ and $nge sk+(s-1)(k-1).$ Some other consequences of our results are presented as well." @default.
- W2497283602 created "2016-08-23" @default.
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- W2497283602 date "2016-07-20" @default.
- W2497283602 modified "2023-09-27" @default.
- W2497283602 title "Two problems of P. Erdős on matchings in set families." @default.
- W2497283602 hasPublicationYear "2016" @default.
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