Size-Ramsey numbers of tight paths

TL;DR

This paper proves that the size-Ramsey number for tight paths in hypergraphs grows linearly with the number of vertices, for any fixed uniformity and number of colours, resolving a previously open question.

Contribution

It establishes the linearity of the size-Ramsey number for hypergraph tight paths, answering a question posed in 2017.

Findings

Size-Ramsey number of hypergraph tight paths is linear in n.

Valid for all fixed r and s, where r is the uniformity and s the number of colours.

Addresses an open problem from Dudek et al. (2017).

Abstract

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r$ and $s$, thereby answering a question of Dudek, La Fleur, Mubayi and R\"odl (2017).