Degree Ramsey numbers for cycles and blowups of trees

Let H→sG mean that every s-coloring of E(H) produces a monochromatic copy of G in some color class. Let the s-color degree Ramsey number of a graph G, written RΔ(G;s), be min{Δ(H):H→sG}. We prove that the 2-color degree Ramsey number is at most 96 for every even cycle and at most 3458 for every odd cycle. For the general s-color problem on even cycles, we prove RΔ(C2m;s)≤16s6 for all m, and RΔ(C4;s)≥0.007s14/9. The constant upper bound forRΔ(Cn;2) uses blowups of graphs, where the d-blowup of a graph G is the graph G′ obtained by replacing each vertex of G with an independent set of size d and each edge e of G with a copy of the complete bipartite graph Kd,d. We also prove the existence of a function f such that if G′ is the d-blowup of G, then RΔ(G′;s)≤f(RΔ(G;s),s,d).