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 K _n 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 K _n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K _t we have c ₁ mt ²/ln t ≤ f(m, K _t) ≤ c ₂mt ², and c′ ₁mt ³/ln t ≤ g(m, K _t) ≤ c′ ₂mt ³/ln t, where c ₁, c ₂, 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.