The size-Ramsey number of 3-uniform tight paths

TL;DR

This paper proves that the size-Ramsey number of 3-uniform tight paths grows linearly with the number of vertices, resolving a previously open question and improving upon earlier bounds.

Contribution

It establishes the linear bound for the size-Ramsey number of 3-uniform tight paths, answering an open problem in hypergraph Ramsey theory.

Findings

Size-Ramsey number of 3-uniform tight paths is linear in n

Improves previous bound from O(n^{3/2} log^{3/2} n) to O(n)

Provides a new approach to hypergraph Ramsey problems

Abstract

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and R\"odl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.