Ramsey numbers upon vertex deletion

IF 0.9 3区 数学 Q2 MATHEMATICS
Yuval Wigderson
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引用次数: 0

Abstract

Given a graph G $G$ , its Ramsey number r ( G ) $r(G)$ is the minimum N $N$ so that every two-coloring of E ( K N ) $E({K}_{N})$ contains a monochromatic copy of G $G$ . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from G $G$ , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs { G n } $\{{G}_{n}\}$ so that in any Ramsey coloring for G n ${G}_{n}$ (i.e., a coloring of a clique on r ( G n ) 1 $r({G}_{n})-1$ vertices with no monochromatic copy of G n ${G}_{n}$ ), one of the color classes has density o ( 1 ) $o(1)$ .

顶点删除后的拉姆齐数
康伦、福克斯和苏达科夫猜想,如果从一个图中删除一个顶点,拉姆齐数最多只能以一个常数因子变化。我们推翻了这一猜想,展示了一个无限图族,从每个图中删除一个顶点都会使拉姆齐数减少一个超常数因子。这一结果的一个结果如下。存在这样一个图族:在任何拉姆齐着色(即没有单色副本的顶点上的一个小群的着色)中,其中一个颜色类的密度为 .
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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