The size Ramsey number of graphs with bounded treewidth

06/21/2019

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by   Nina Kamcev, et al.

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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.