Unavoidable patterns

Abstract: Let denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every and positive integer k there is such that every 2-edge-coloring of the complete graph of order which has at least edges in each color contains a member of . This conjecture was proved by Cutler and Montágh, who showed that . We give a much simpler proof of this conjecture which in addition shows that for some constant c. This bound is tight up to the constant factor in the exponent for all k and ϵ. We also discuss similar results for tournaments and hypergraphs. [Copyright &y& Elsevier]