# Ramsey properties of randomly perturbed graphs: cliques and cycles

## TL;DR

This paper investigates the Ramsey properties of randomly perturbed graphs, determining the minimum random edges needed to ensure certain monochromatic subgraphs appear, extending previous results to new graph pairs and cycles.

## Contribution

It provides precise thresholds for Ramsey properties in randomly perturbed graphs for cliques and cycles, generalizing earlier work and resolving several open problems.

## Key findings

- Established thresholds for (K_s,K_t)-Ramsey in perturbed graphs for s,t ≥ 5.
- Resolved the (K_4,K_4)-Ramsey problem with new bounds and constructions.
- Extended results to (C_s,C_t)-Ramsey and mixed cycle-clique scenarios.

## Abstract

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 $t\ge 4$).