Degree Ramsey numbers for cycles and blowups of trees

Let H ->(s) G 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 R-Delta(G; s), be min{Delta(H): H ->(s) G}. 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 R-Delta(C-2m; s) <= 16s(6) for all m, and R-Delta (C-4; s) >= 0.007s(14/9). The constant upper bound for R-Delta(C-n; 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 K-d.d. We also prove the existence of a function f such that if G' is the d-blowup of G, then R-Delta(G'; s) <= f (R-Delta(G; s), s, d). (c) 2012 Elsevier Ltd. All rights reserved.