$r(4,t)$ 的渐近线 | 数学年鉴

IF 5.7 1区数学 Q1 MATHEMATICS

Annals of Mathematics 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

Annals of Mathematics 数学-数学

CiteScore

9.10

自引率

2.00%

发文量

审稿时长

12 months

期刊介绍： The Annals of Mathematics is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Founded in 1884 by Ormond Stone of the University of Virginia, the journal was transferred in 1899 to Harvard University, and in 1911 to Princeton University. Since 1933, the Annals has been edited jointly by Princeton University and the Institute for Advanced Study.