The minimum number of subgraphs in a graph and its complement

For a graphb 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 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 . 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 .