Degree Ramsey numbers of closed blowups of trees (Q405189)

Summary: The degree Ramsey number of a graph \(G\), denoted \(R_\Delta(G;s)\), is \(\min\{\Delta(H): H\overset {s}{\rightarrow} G\}\), where \(H\overset {s}{\rightarrow} G\) 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 \(R_\Delta(G;s) \leq f(\Delta(G), s)\) when \(G\) is a closed blowup of a tree.