Edge-Colorings of Complete Graphs that Avoid Polychromatic Trees

Given a positive integer n and a family F of graphs, let R∗(n, F) denote the maximum number of colors in an edge-coloring of Kn such that no subgraph of Kn belonging to F has distinct colors on its edges. We determine R∗(n, Tk) where Tk is the family of trees with k edges. We derive general bounds for R∗(n, T) where T is an arbitrary tree with k edges. Finally, we present a single tree T with k edges such that R∗(n, T) is nearly as small as R∗(n, Tk).