Asymmetric Ramsey properties of random graphs involving cliques and cycles

We prove that for every ℓ, r ≥ 3, there exists c > 0 such that for (image found), with high probability there is a 2-edge-colouring of the random graph Gn,p with no monochromatic copy of Kr of the first colour and no monochromatic copy of Cℓ of the second colour. This is a progress on a conjecture of Kohayakawa and Kreuter

Ramsey properties of randomly perturbed graphs: cliques and cycles

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

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

Algorithmic and enumerative aspects of the Moser-Tardos distribution

Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during intermediate stages of MT. We show that these configurations have a more or less "random" form, building further on the "MT-distribution" concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving upon MT across the complexity spectrum: it makes some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial run-times in some settings, transforms certain super-polynomial-time algorithms into polynomial-time ones, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known. We show that in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution which avoids most of the bad events. We show in some cases this MT variant can run faster than the original MT algorithm itself, and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals -- improving upon earlier bounds of Stein (1975) -- among other applications. We furthermore give applications in enumeration, showing that most applications (where we aim for all or most of the bad events to be avoided) have many more solutions than known before by proving that the MT-distribution has "large" min-entropy and hence that its support-size is large

An approximate version of Sidorenko's conjecture

A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page