THE MINIMUM NUMBER OF SUBGRAPHS IN A GRAPH AND ITS COMPLEMENT

For a graph F without isolated vertices, let M(F;n) denote the minimum number of monochromatic copies of F in any 2-coloring of the edges of K(n). Burr and Rosta conjectured that M(F;n) = n(t)/2u-1a + O(n(t-1)) when F has order t, size u, and a automorphisms. Independently, Sidorenko and Thomason have shown that the conjecture is false. We give families of graphs F of order t, of size u, and with a automorphisms where (M(F;n)2u-1a/n(t) --> 0 as t --> infinity. We show also that the asymptotic value of M(F;n) is not solely a function of the order, size, and number of automorphisms of F.