非对角与三部分典型拉姆齐数的锐界

中文摘要 AI 辅助

典型拉姆齐定理确立了：对于每个正整数 $t$，存在足够大的 $n$，使得完全图 $K_n$ 的每条边着色都包含一个典型着色的 $K_t$ 副本，即单色、彩虹或字典序着色。本文研究该定理的两个变体。首先，我们在多部分超图背景下研究典型拉姆齐数。我们证明：在至少 $t^{151t^2}$ 个顶点的 3-一致完全超图的每条边着色中，总是存在一个典型着色的 $K_{t, t, t}^{(3)}$ 副本。该估计在指数常数意义下是锐的。其次，我们探索非对角典型拉姆齐数。设 $ER(a, b, c)$ 表示保证完全图 $K_n$ 的每条边着色中必有单色 $K_a$、字典序 $K_b$ 或彩虹 $K_c$ 所需的最小顶点数 $n$。我们在不同参数区间内建立了这些数的锐界。具体而言，我们证明当 $c$ 相对于 $a$ 足够大时，$ER(a, b, c)\geq c^{\Omega(ab)}$；当 $b\ge 4$ 为固定常数且 $c\le a$ 时，$ER(a, b, c)\leq 2^{O_b(a)}$。最后，我们分析避免字典序三角形（即 $b=3$）时该函数的行为，证明 $ER(a, 3, c)\le (a-1)(c-2)+O(c^6)$。

英文摘要

The canonical Ramsey theorem establishes that for every positive integer $t$, there is a sufficiently large $n$ such that every edge-colouring of a complete graph $K_n$ contains a copy of $K_t$ that is canonically coloured, i.e. monochromatic, rainbow, or lexicographic. In this paper we investigate two variations of this theorem. First, we study canonical Ramsey numbers in the multipartite hypergraph setting. We prove that in every edge-colouring of the 3-uniform complete hypergraph on at least $t^{151t^2}$ vertices, there always exists a canonically coloured copy of $K_{t, t, t}^{(3)}$. This estimate is sharp up to the constant in the exponent. Second, we explore off-diagonal canonical Ramsey numbers. Let $ER(a, b, c)$ denote the minimum number of vertices $n$ required to guarantee a monochromatic $K_a$, a lexicographic $K_b$, or a rainbow $K_c$ in every edge-colouring of the complete graph $K_n$. We establish sharp bounds for these numbers across different parameter regimes. Specifically, we prove that $ER(a, b, c)\geq c^{Ω(ab)}$ when $c$ is sufficiently large relative to $a$, and that $ER(a, b, c)\leq 2^{O_b(a)}$ when $b\ge 4$ is a fixed constant and $c\le a$. Finally, we analyse the behaviour of the function when avoiding lexicographic triangles (i.e. $b=3$), showing that $ER(a, 3, c)\le (a-1)(c-2)+O(c^6)$.

发表机构

• ETH Zurich（苏黎世联邦理工学院）

机构由 AI 辅助整理，请以论文原文为准。

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17 pages, including references