A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that “every edge coloured graph of a certain type contains a rainbow matching using every colour.” In this paper we introduce a versatile “sampling trick,” which allows us to asymptotically solve some well-known conjectures and to obtain short proofs of old results. In particular: $\bullet $ We give the first asymptotic proof of the “non-bipartite” Aharoni–Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky, and Zerbib. $\bullet $ We give a very short asymptotic proof of Grinblat’s conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat’s problem as a function of edge multiplicity of the corresponding multigraph. $\bullet $ We give the first asymptotic proof of a 30-year-old conjecture of Alspach. $\bullet $ We give a simple proof of Pokrovskiy’s asymptotic version of the Aharoni–Berger conjecture with greatly improved error term.

中文翻译：

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彩虹匹配结果的简短证明

如果边缘颜色图的所有边缘都具有不同的颜色，则该子图称为彩虹。彩虹子图的研究可以追溯到欧拉在拉丁方阵上的工作，此后一直是广泛研究的焦点。该领域的许多猜想粗略地说“某种类型的每个边缘彩色图都包含使用每种颜色的彩虹匹配”。在本文中，我们介绍了一种通用的“采样技巧”，它使我们能够渐近地解决一些众所周知的猜想并获得旧结果的简短证明。特别是： $\bullet $ 我们给出了“非二分”Aharoni-Berger 猜想的第一个渐近证明，解决了 Aharoni、Berger、Chudnovsky 和 ​​Zerbib 的两个猜想。$\bullet $ 我们给出了 Grinblat 猜想的一个非常简短的渐近证明（首先由 Clemens、Ehrenmüller 和 Pokrovskiy 获得）。此外，我们为 Grinblat 问题获得了一个新的渐近紧界，作为相应多重图的边多重性的函数。$\bullet $ 我们给出了一个 30 年前的 Alspach 猜想的第一个渐近证明。$\bullet $ 我们给出了一个简单的证明，证明了 Pokrovskiy 的 Aharoni-Berger 猜想的渐近版本，误差项大大改进。