REPEATED PATTERNS in PROPER COLORINGS

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 Kn 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 Kn with at most cn² colors contains two repeats of T, while there are colorings with at most c^\prime n³/² colors for some absolute constant c^\prime 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 Kn 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.