Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of Kn, yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edge-coloring of Kn yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = Kt, we have c1mt2/ln t ≤ f(m, Kt) ≤ c2mt2, and c'1mt3/ln t ≤ g(m, Kt) ≤ c'2mt3/ln t, where c1, c2, c'1, c'2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d.