Satisfying sequences for rainbow partite matchings

Let ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_s\subset [n{]}^k$ be a collection of $s$ families. In this paper, we address the following question: for which sequences $f_1,\dots ,f_s$ the conditions $\mathrm{\mid }{\text{ff}}_i\mathrm{\mid }>f_i$ imply that the families contain a rainbow matching, that is, there are pairwise disjoint $F_1\in {\text{ff}}_1,\dots F_s\in {\text{ff}}_s$? We call such sequences {\em satisfying}. Kiselev and the first author verified the conjecture of Aharoni and Howard and showed that $f_1=\dots =f_s=(s-1)n^{k-1}$ 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.