Unavoidable patterns

Let Fk 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 2 > 0 and positive integer k there is an n(k, 2) such that every 2-edge-coloring of the complete graph of order n ≥ n(k, 2) which has at least 2n2 ) edges in each color contains a member of Fk. This conjecture was proved by Cutler and Montágh, who showed that n(k, 2) < 4. We give a much simpler proof of this conjecture which in addition shows that n(k, 2) < 2−ck for some constant c. This bound is tight up to the constant factor in the exponent for all k and 2. We also discuss similar results for tournaments and hypergraphs.