The Size Ramsey Number of Graphs with Bounded Treewidth

Abstract

A graph G is Ramsey for a graph H if every 2-coloring 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 in general 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.

The size‐Ramsey number of powers of bounded degree trees

“…While finalising this paper, we learned that Kamčev et al . [19] proved, among other things, that the 2‐colour size‐Ramsey number of an ‐vertex graph with bounded degree and bounded treewidth is . This is equivalent to our result for .…”

“…Conversely, bounded powers of bounded degree trees have bounded treewidth and bounded degree. Therefore, we obtain the following equivalent version of Theorem 1, which generalises the result from [19] and answers one of their main open questions ([19, Question 5.2]). Corollary For any positive integers , and and any ‐vertex graph with treewidth and , we have …”

The size-Ramsey number of cubic graphs

“…Subsequent work has extended this result to many other familes of graphs, including boundeddegree trees [14], cycles [20], and, more recently, powers of paths and bounded-degree trees [5] and long subdivisions [11]. For some further recent developments, see [7,18,19,22,25].…”

Three early problems on size Ramsey numbers

“…The fourth question, about the size Ramsey number of paths, was resolved by Beck [2], who proved the surprising result that r(P n ) = Θ(n) for the path P n with n vertices. This breakthrough inspired many of the subsequent developments in the field, such as the classic papers [3,17,20,23,28] and the more recent results in [4,5,6,12,13,14,18,19,22,25].…”