Off-diagonal book Ramsey numbers

IF 0.9 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Combinatorics, Probability & Computing Pub Date : 2023-01-09 DOI:10.1017/s0963548322000360

David Conlon, Jacob Fox, Yuval Wigderson

引用次数: 6

Abstract

Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ .

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非对角线书拉姆齐号码

图书图$B_n ^{(k)}$由$K_{k+1}$沿公共$K_k$连接的$n$个副本组成。在本文的前传中，我们研究了对角拉姆齐数$r(B_n ^{(k)}， B_n ^{(k)})$。这里我们考虑自然非对角线变量$r(B_{cn} ^{(k)}， B_n^{(k)})$对于固定$c \in(0,1]$。在这个更一般的设置中，我们展示了一个有趣的二分法:对于非常小的$c$，一个简单的$k$ -部构造决定了Ramsey函数和所有近极值着色都接近于$k$ -部，而对于$c$远离$0$，适当密度的随机着色是渐近最优的，所有近极值着色都是拟随机的。我们的调查还提出了一系列关于中间值c的问题。

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

Combinatorics, Probability & Computing 数学-计算机：理论方法

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.