Degree Ramsey Numbers of Closed Blowups of Trees

The degree Ramsey number of a graph G, denoted R∆(G; s), is min{∆(H) : H s → G}, where H s → G means that every s-edge-coloring of H contains a monochromatic copy of G. The closed k-blowup of a graph is obtained by replacing every vertex with a clique of size k and every edge with a complete bipartite graph where both partite sets have size k. We prove that there is a function f such that R∆(G; s) 6 f(∆(G), s) when G is a closed blowup of a tree.