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\documentclass{article}\pagestyle{empty}\begin{document}$$ M(F;n) = \frac{{n^t }}{{2^{u - 1} a}} + O(n^{t - 1}) $$\end{document}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\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{M(F;n)2^{u - 1} a}}{{n^t }} \to 0\,{\rm as}\,t \to \infty $$\end{document} . 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 . © 1929 John Wiley & Sons, Inc.