Edge-colorings of complete graphs that avoid polychromatic trees

AbstractGiven 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)