Unavoidable patterns

Let F k 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 ϵ > 0 and positive integer k there is n ( k , ϵ ) such that every 2-edge-coloring of the complete graph of order n ⩾ n ( k , ϵ ) which has at least ϵ ( n 2 ) edges in each color contains a member of F k . This conjecture was proved by Cutler and Montágh, who showed that n ( k , ϵ ) < 4 k / ϵ . We give a much simpler proof of this conjecture which in addition shows that n ( k , ϵ ) < ϵ − c k 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.