# The size Ramsey number of graphs with bounded treewidth

## TL;DR

This paper investigates the size Ramsey number for graphs with bounded treewidth, showing that such graphs can be made sparse under certain conditions, but their treewidth cannot be universally bounded in Ramsey graphs.

## Contribution

It establishes bounds on the size and treewidth of Ramsey graphs for graphs with bounded treewidth, extending previous results to a broader class of graphs.

## Key findings

- Sparse Ramsey graphs exist for graphs with bounded degree and treewidth.
- Treewidth of Ramsey graphs cannot be bounded solely by the treewidth of the target graph.
- Results extend known bounds from bounded bandwidth to bounded treewidth.

## Abstract

A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree.