The size-Ramsey number of powers of paths

TL;DR

This paper proves that the size-Ramsey number of powers of paths grows linearly with the number of vertices, answering a previously open question and providing a probabilistic, yet constructive, proof.

Contribution

It establishes that for any fixed power k, the size-Ramsey number of the k-th power of a path is linear in the number of vertices, solving an open problem.

Findings

Size-Ramsey number of path powers is O(n)

Proof uses probabilistic methods, adaptable to constructive approaches

Answers a question posed by Conlon

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.