On the size-Ramsey number of grids

TL;DR

This paper establishes an improved upper bound on the size-Ramsey number of grid graphs, demonstrating that it grows at most as fast as O(n^{5/4}), which advances previous bounds significantly.

Contribution

The authors provide a tighter upper bound on the size-Ramsey number of grid graphs, improving upon earlier results by refining the growth rate to O(n^{5/4}).

Findings

Size-Ramsey number of the grid is O(n^{5/4})

Improves previous bound of n^{3/2 + o(1)}

Advances understanding of Ramsey properties of grid graphs

Abstract

We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.