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 Δ ( G ; s ) , be min { Δ ( 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 Δ ( C 2 m ; s ) ≤ 16 s 6 for all m , and R Δ ( C 4 ; s ) ≥ 0.007 s 14 / 9 . The constant upper bound for R Δ ( 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 Δ ( G ′ ; s ) ≤ f ( R Δ ( G ; s ) , s , d ) .