Turán theorems for unavoidable patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and Fox Sudakov. First, t ≥ 3, we show that any two-colouring of the complete graph on n vertices is δ -far from being monochromatic contains an unavoidable t-colouring when ≫ −1/ , where -colouring a clique order 2 in which one colour forms either or disjoint cliques . Next, tournament transitive t-tournament −1/[ /2] -tournament blow-up cyclic triangle obtained replacing each vertex Conditional well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants hence determine magnitude corresponding off-diagonal numbers.