Ramsey properties of randomly perturbed graphs: cliques and cycles

Abstract

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.

Random Perturbation of Sparse Graphs

“…In most of these cases a G α , that is responsible for the lower bound, is the complete imbalanced bipartite graph K αn,(1−α)n . In this model there are also results with lower bounds on α [6,16,23,38] and for Ramsey-type problems [13,14].…”

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

“…In recent years, a range of results have been obtained concerning embedding spanning subgraphs into a randomly perturbed graph, as well as other properties of the model; see, for example, [2,3,5,6,8,9,13,14,26,30,35,36]. The model has also been investigated in the setting of directed graphs and hypergraphs (see e.g., [4,23,34,41]).…”

Vertex Ramsey properties of randomly perturbed graphs

“…In 2006, Krivelevich, Sudakov and Tetali [19] initiated the study of (edge) Ramsey properties of randomly perturbed graphs. Combining their work with recent results of the first and third authors [7] and of Powierski [23], we now know the number of random edges one needs to add to an n-vertex graph of positive density to w.h.p. ensure the resulting graph is (K r , K s )-Ramsey for all values of (r, s), except for the case when r = 4 and s ≥ 5.…”

“…ensure the resulting graph is ( K r , K s )‐Ramsey for all values of ( r , s ), except for the case when r = 4 and s ≥ 5. See 7 for other results on this topic.…”