Degree Ramsey numbers for even cycles

Let H s − → G denote that any s-coloring of E(H) contains a monochromatic G. The degree Ramsey number of a graph G, denoted by R∆(G, s), is min{∆(H) : H s − → G}. We consider degree Ramsey numbers where G is a fixed even cycle. Kinnersley, Milans, and West showed that R∆(C2k, s) ≥ 2s, and Kang and Perarnau showed that R∆(C4, s) = Θ(s 2). Our main result is that R∆(C6, s) = Θ(s 3/2) and R∆(C10, s) = Θ(s 5/4). Additionally, we substantially improve the lower bound for R∆(C2k, s) for general k.