Abstract
For a fixed graph H, what is the smallest number of colors C such that there is a proper edge-coloring of the complete graph K-n with C colors containing no two vertex-disjoint color-isomorphic copies, or repeats, of H? We study this function and its generalization to more than two copies using a variety of combinatorial, probabilistic, and algebraic techniques. For example, we show that for any tree T there exists a constant c such that any proper edge-coloring of K-n with at most cn(2) colors contains two repeats of T, while there are colorings with at most c'n(3/2) colors for some absolute constant c' containing no three repeats of any tree with at least two edges.
We also show that for any graph H containing a cycle there exist k and c such that there is a proper edge-coloring of K-n with at most cn colors containing no k repeats of H, while for a tree T with m edges, a coloring with o(n((m+1)/m)) colors contains omega(1) repeats of T.