The size-Ramsey number of powers of paths

Dennis Clemens; Matthew Jenssen; Yoshiharu Kohayakawa; Natasha Morrison; Guilherme Oliveira Mota; Damian Reding; Barnaby Roberts

arXiv · 1707.04297

The size-Ramsey number of powers of paths

Dennis Clemens·Matthew Jenssen·Yoshiharu Kohayakawa·Natasha Morrison·Guilherme Oliveira Mota·Damian Reding·Barnaby Roberts

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Abstract

Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is $q$-Ramsey for $H$, denoted $G\rightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is defined to be $\hat{r}(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}$. Answering a question of Conlon, we prove that, for every fixed $k$, we have $\hat{r}(P_n^k)=O(n)$, where $P_n^k$ is the $k$-th power of the $n$-vertex path $P_n$ (i.e. , the graph with vertex set $V(P_n)$ and all edges $\{u,v\}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.

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Dennis Clemens, Matthew Jenssen, Yoshiharu Kohayakawa, Natasha Morrison, Guilherme Oliveira Mota, Damian Reding, Barnaby Roberts. 2017-07-13. The size-Ramsey number of powers of paths. https://arxiv.org/abs/1707.04297

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