# On the size-Ramsey number of grid graphs

## TL;DR

This paper establishes an upper bound of approximately n^{3} for the size-Ramsey number of grid graphs, advancing understanding of edge-minimal graphs that guarantee monochromatic grid copies under any 2-coloring.

## Contribution

It provides a new upper bound on the size-Ramsey number for grid graphs, improving previous results and deepening insight into their Ramsey properties.

## Key findings

- Size-Ramsey number of n×n grid graphs is at most n^{3+o(1)}.
- The result narrows the gap in understanding the minimal edge count for grid graph Ramsey properties.
- Advances theoretical bounds in combinatorial graph theory.

## Abstract

The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that the size-Ramsey number of the grid graph on $n\times n$ vertices is bounded from above by $n^{3+o(1)}$.