$r(4,t)$ 的渐近线 | 数学年鉴

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

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引用次数: 0

摘要

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。

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

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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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

期刊介绍： ACS Applied Materials & Interfaces is a leading interdisciplinary journal that brings together chemists, engineers, physicists, and biologists to explore the development and utilization of newly-discovered materials and interfacial processes for specific applications. Our journal has experienced remarkable growth since its establishment in 2009, both in terms of the number of articles published and the impact of the research showcased. We are proud to foster a truly global community, with the majority of published articles originating from outside the United States, reflecting the rapid growth of applied research worldwide.