Ramsey properties of randomly perturbed graphs: cliques and cycles

Given graphs H₁, H₂, a graph G is (H₁, H₂) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H₁ or a blue copy of H₂. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K₃, K_t) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K₃, K_t). We make significant progress on this question, giving a precise solution in the case when H₁ = K_s and H₂ = K_t where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K₃, K_t) -Ramsey question. Moreover, we give bounds for the corresponding (K₄, K_t) -Ramsey question; together with a construction of Powierski [37] this resolves the (K₄, K₄) -Ramsey problem.

We also give a precise solution to the analogous question in the case when both H₁ = C_s and H₂ = C_t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (C_s, K_t) -Ramsey (for odd s ≽ 5 and t ≽ 4).

To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.