The asymptotics of $r(4,t)$ | Annals of Mathematics

IF 8.3 2区材料科学 Q1 MATERIALS SCIENCE, MULTIDISCIPLINARY

ACS Applied Materials & Interfaces Pub Date : 2024-03-05 DOI:10.4007/annals.2024.199.2.8

Sam Mattheus, Jacques Verstraete

引用次数: 0

Abstract

For integers $s,t \geq 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = \Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty,\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erdős.

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$r(4,t)$ 的渐近线 | 数学年鉴

对于整数 $s,t \geq 2$，拉姆齐数 $r(s,t)$ 表示每个 $n$ 顶点图包含一个阶为 $s$ 的簇或一个阶为 $t$ 的独立集的最小值 $n$。在本文中，我们证明了[ r(4,t) = \Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty,\] 这决定了 $r(4,t)$ 直到一个阶为 $\log^2 \! t$ 的因子，并解决了厄多斯的一个猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Materials & Interfaces 工程技术-材料科学：综合

CiteScore

16.00

自引率

6.30%

发文量

4978

审稿时长

1.8 months

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