Extremal and Ramsey results on graph blowups

IF 0.4 Q4 MATHEMATICS, APPLIED Journal of Combinatorics Pub Date : 2019-12-18 DOI:10.4310/joc.2021.v12.n1.a1
J. Fox, Sammy Luo, Yuval Wigderson
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引用次数: 3

Abstract

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.
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图放大的极值和拉姆齐结果
最近,Souza引入了放大拉姆齐数作为二部拉姆齐数的推广。对于图$G$和$H$,如果$G$的每个$r$边着色包含$H$的单色副本,则称$G\overset{r}{\ lonightarrow} H$。设$H[t]$表示$H$的$t$放大。然后将$G,H,r,$和$t$的放大拉姆齐数定义为使$G[n] \覆盖{r}{\ longightarrow} H[t]$的最小值$n$。Souza证明了$n$的上界和下界是$t$的指数,并推测指数常数不依赖于$G$。我们证明了指数常数中对$G$的依赖确实是不必要的,但我们猜想对$G$的依赖是不可避免的。在Souza的证明和我们的证明中,一个重要的步骤是Nikiforov的定理,它说如果一个图包含H$的可能副本的常数部分,那么它包含一个对数大小的H$的放大。我们还提供了一个新的证明,它具有更好的数量依赖性。
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来源期刊
Journal of Combinatorics
Journal of Combinatorics MATHEMATICS, APPLIED-
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发文量
21
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