Degree Ramsey Numbers of Graphs

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∆(G; s), be min{∆(H) : H s → G}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then R∆(T ; s) = s(k− 1) + ǫ, where ǫ = 1 when k is odd and ǫ = 0 when k is even. For general trees, R∆(T ; s) ≤ 2s(∆(T ) − 1). To study sharpness of the upper bound, consider the double-star Sa,b, the tree whose two non-leaf vertices have have degrees a and b. If a ≤ b, then R∆(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then R∆(Sb,b; s) = f(s)(b− 1) − o(b), where f(s) = 2s− 3.5 −O(s−1). We prove several results about edge-colorings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that R∆(C2k+1; s) ≥ 2s + 1, that R∆(C2k; s) ≥ 2s, and that R∆(C4; 2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edgecoloring such that the subgraph in each color class has girth at least 5.