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 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 edges in each color contains a member of Fk. This conjecture was proved by Cutler and Montágh, who showed that n(k,ϵ)<4k/ϵ. We give a much simpler proof of this conjecture which in addition shows that n(k,ϵ)<ϵ−ck 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.