Degree Ramsey numbers for even cycles

Let H⟶sG denote that any s-coloring of E(H) contains a monochromatic G. The degree Ramsey number of a graph G, denoted by RΔ(G,s), is min{Δ(H):H⟶sG}. We consider degree Ramsey numbers where G is a fixed even cycle. Kinnersley, Milans, and West showed that RΔ(C2k,s)≥2s, and Kang and Perarnau showed that RΔ(C4,s)=Θ(s2). Our main result is that RΔ(C6,s)=Θ(s3∕2) and RΔ(C10,s)=Θ(s5∕4). Additionally, we substantially improve the lower bound for RΔ(C2k,s) for general k.