Let $\mathcal F_1,\ldots, \mathcal F_s\subset [n]^k$ be a collection of $s$ families. In this paper, we address the following question: for which sequences $f_1,\ldots, f_s$ the conditions $|\ff_i|>f_i$ imply that the families contain a rainbow matching, that is, there are pairwise disjoint $F_1\in \ff_1,\ldots F_s\in \ff_s$? We call such sequences {\em satisfying}. Kiselev and the first author verified the conjecture of Aharoni and Howard and showed that $f_1 = \ldots = f_s=(s-1)n^{k-1}$ is satisfying for $s>470$. This is the best possible if the restriction is uniform over all families. However, it turns out that much more can be said about asymmetric restrictions. In this paper, we investigate this question in several regimes and in particular answer the questions asked by Kiselev and Kupavskii. We use a variety of methods, including concentration and anticoncentration results, spread approximations, and Combinatorial Nullstellenzats.

翻译：设 $\mathcal F_1,\ldots, \mathcal F_s\subset [n]^k$ 为 $s$ 个族的集合。本文研究以下问题：对于哪些序列 $f_1,\ldots, f_s$，条件 $|\ff_i|>f_i$ 能推出这些族包含一个彩虹匹配，即存在两两不交的 $F_1\in \ff_1,\ldots F_s\in \ff_s$？我们称这样的序列为{\em 满足序列}。Kiselev 与第一作者验证了 Aharoni 和 Howard 的猜想，并证明了当 $s>470$ 时，$f_1 = \ldots = f_s=(s-1)n^{k-1}$ 是满足序列。若限制对所有族是均匀的，则此结果是最优的。然而，对于非对称限制，实际上可以得到更强的结论。本文在多种情形下研究该问题，并特别回答了 Kiselev 与 Kupavskii 提出的问题。我们运用了多种方法，包括集中与反集中结果、展开逼近以及组合零点定理。