Unavoidable patterns

Let \mathcal{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. Bollob\'as conjectured that for every \epsilon>0 and positive integer k there is an n(k,\epsilon) such that every 2-edge-coloring of the complete graph of order n \geq n(k,\epsilon) which has at least \epsilon {n \choose 2} edges in each color contains a member of \mathcal{F}_k. This conjecture was proved by Cutler and Mont\'agh, who showed that n(k,\epsilon)<4^{k/\epsilon}. We give a much simpler proof of this conjecture which in addition shows that n(k,\epsilon)<\epsilon^{-ck} for some constant c. This bound is tight up to the constant factor in the exponent for all k and \epsilon. We also discuss similar results for tournaments and hypergraphs.