On the number of edges not covered by monochromatic copies of a fixed graph

For a fixed graph H; let f ðn;HÞ denote the maximum possible number of edges not belonging to a monochromatic copy of H in a 2-edge-coloring of the complete graph of order n: Let exðn;HÞ be the Turán number of H; i.e., the maximum number of edges that a graph on n vertices can have without containing a copy of H: An easy lower bound of f ðn;HÞXexðn;HÞ follows from the 2-edge-coloring in which the edges of one color form the largest H-free graph. In this paper we consider the cases when H is an edge-color-critical graph (e.g., a complete graph) or a 4-cycle. We will show then, that for sufficiently large n; the value of f ðn;HÞ is actually equal to exðn;HÞ: r 2003 Elsevier Inc. All rights reserved.