Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every
colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$
or a blue copy of $H_2$. 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 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 in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability $(K_3,K_t)$-Ramsey (for
$t\ge 3$). They also raised the question of generalising this result to pairs
of graphs other than $(K_3,K_t)$. We make significant progress on this
question, giving a precise solution in the case when $H_1=K_s$ and $H_2=K_t$
where $s,t \ge 5$. 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 $(K_3,K_t)$-Ramsey question. Moreover, we
give bounds for the corresponding $(K_4,K_t)$-Ramsey question; together with a
construction of Powierski this resolves the $(K_4,K_4)$-Ramsey problem.
  We also give a precise solution to the analogous question in the case when
both $H_1=C_s$ and $H_2=C_t$ are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali 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 $s\ge 5$
and $t\ge 4$).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
  Powierski, stated results for cliques in graphs of low positive density
  separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
  appear in CP

Das, Shagnik

Treglown, Andrew

English

arXiv

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, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt 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, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -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 = Cs and H2 = Ct 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 (Cs, Kt) -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

Ramsey properties of randomly perturbed graphs: cliques and cycles

Abstract

Similar works

Full text

Available Versions

University of Birmingham Research Portal

University of Birmingham Research Portal