Satisfying sequences for rainbow partite matchings

TL;DR

The paper advances the study of asymmetric thresholds for rainbow cross-matchings in $k$-partite hypergraphs by developing a multi-faceted toolkit that blends spread approximations, concentration and anticoncentration analysis, and algebraic methods. It establishes that the arithmetic progression sequence $(f_i)=(i n^{k-1})$ is satisfying in broad regimes of $n$ (via spread-approximation techniques) and analyzes a truncated progression to map the boundary where asymmetry ceases to guarantee a rainbow matching (using anticoncentration structure). For $k=2$ and prime $n$, it further provides a Nullstellensatz-based proof that many natural sequences are satisfying. Collectively, these results illuminate the landscape of asymmetric thresholds and demonstrate how structural anti-concentration phenomena underpin existence of rainbow matchings in multipartite hypergraphs.

Abstract

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.

Theorems & Definitions (37)

• Conjecture 1: KKconc
• Theorem 1
• Theorem 2
• Theorem 3
• Claim 4
• proof
• Claim 5
• proof
• Claim 6
• proof