The sharp threshold for rainbow stackings of random edge-colourings

IF 1 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2026-10-07 DOI:10.1112/blms.70522

Hong Liu, Guorui Ma, Yangrui Xiang, Zhifei Yan

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Abstract

A rainbow stacking of $m$ independent, uniformly random $r$ -edge-colourings of $K_{n}$ is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette size $r$ required for the existence of such stackings was recently initiated by Alon, Defant and Kravitz (Bull. Lond. Math. Soc. 57 (2025), 1656–1670), who bounded the phase transition within a constant-order window around $\frac{m (\frac{n}{2})}{2 \log (n!)}$ . We determine the constant term in this transition. For every fixed $m ⩾ 2$ and every function $ω (n) \to \infty$ , with high probability there is no rainbow stacking if

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随机边缘颜色的彩虹堆叠的尖锐阈值

一个由m个$m$独立的、均匀随机的r $r$ - K n $K_n$的边着色组成的彩虹叠加是一个顶点排列的元组，它叠加了这些着色，使得没有两个相同颜色的边重叠。最近，Alon， Defant和Kravitz （Bull）发起了对这种叠层存在所需的临界调色板尺寸r $r$的研究。很长。数学。Soc. 57 (2025), 1656-1670)，他把相变限定在一个常阶窗口内，大约是mn22 - 2log （n !）$\frac{m\binom{n}{2}}{2\log (n!)}$。我们确定了这个跃迁中的常数项。对于每个固定m大于或等于2 $m\geqslant 2$和每个函数ω (n)→∞$\omega (n)\rightarrow \infty$，很可能没有彩虹叠加

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