The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres, proved in 1935.

The r-colour Ramsey number R_r(k) is the minimum n ∈ℕ such that every r-colouring of the edges of the complete graph K_n on n vertices contains a monochromatic copy of K_k. We prove, for each fixed r ⩾ 2, that R_r(k) ⩽ e^-δk r^rk for some constant δ= δ(r) > 0 and all sufficiently large k ∈ℕ. For each r ⩾ 3, this is the first exponential improvement over the upper bound of Erdős and Szekeres from 1935. In the case r = 2, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.

We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi random graph G(n, p). Recently Acan and Kahn showed that the largest such family contains only O(n^2/(logn)^3) cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, Ω (n^2/(logn)^3) , by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound O(n^2/(logn)^3) and discuss the problem of the precise size of the largest such clique packing.

Let N triangle(G)be the number of triangles in a graph G. In Gold-schmidt et al.(2020) and Neeman et al.(2023)(respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdos-Renyi random graphs G(m)similar to G (n, m): P(N-triangle(G((m)))<(1-delta)E[N-triangle(G((m)))]) ={exp(-Theta(delta(2)n(3)))ifn-3/2 <

We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random graph is not too sparse, specifically when the edge probability p=p(n) satisfies p(1-p)n \gg \log n. The problem is more difficult in the sparser regime 1 \ll pn \ll \log n, where we determine the asymptotics of M_n for specific classes of trees. Interestingly, here our large deviation type optimization arguments reveal that the behavior of M_n changes as we vary p=p(n), due to different mechanisms that can make the maximum large.

Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $\mathcal{H}$ satisfies some regularity condition we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the cyclic group $\mathbb{Z}_N$. Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$.

Position n points uniformly at random in the unit square S, and consider the Voronoi tessellation of S corresponding to the set η of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let HS denote the event that there exists a red horizontal crossing of S in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event HS will occur or not. More precisely, since HS occurs with probability 1/2, by symmetry, they conjectured that the conditional probabilities P(HS|η) converge in probability to 1/2, as n → ∞. This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris and Tassion. In this paper we derive a stronger bound on the rate at which P(HS|η) approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.

We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for $p\neq 1/2$, and that every site will change colour infinitely often almost surely for the balanced initial condition $p=1/2$.

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph G ( n , m ) G(n,m) . Our approach is based on applying Freedman’s inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related G ( n , p ) G(n,p) model.

The areas of Ramsey theory and random graphs have been closely linked ever since Erdos' famous proof in 1947 that the 'diagonal' Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the 'off-diagonal' Ramsey numbers R(3, k). In this model, edges of K-n are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted G(n)(,Delta). In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3, k) = Theta (k(2)/log k). In this paper we improve the results of both Bohman and Kim, and follow the triangle-free process all the way to its asymptotic end. In particular, we shall prove that e(G(n,Delta)) = (1/2 root 2 + o(1)) n(3/2) root log n, with high probability as n -> infinity. We also obtain several pseudorandom properties of G(n,Delta), and use them to bound its independence number, which gives as an immediate corollary R(3, k) >= (1\4 - o(1)) k(2)/log k. This significantly improves Kim's lower bound, and is within a factor of 4 + o(1) of the best known upper bound, proved by Shearer over 25 years ago.

We consider a dynamical process on a graph G, in which vertices are infected (randomly) at a rate which depends on the number of their neighbors that are already infected. This model includes bootstrap percolation and first‐passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph , significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph $G(n,m)$. Our approach is based on applying Freedman's inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related $G(n,p)$ model.

We study survival among two competing types in two settings: a planar growth model related to two-neighbour bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncoloured sites are given a colour at rate $0$, $1$ or $\infty$, depending on whether they have zero, one, or at least two neighbours of that colour. In the urn scheme, each vertex of a graph $G$ has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same colour is added to each neighbouring urn, and balls in the same urn but of different colours annihilate on a one-for-one basis. We show that, for every connected graph $G$ and every initial configuration, only one colour survives almost surely. As a corollary, we deduce that in the two-type growth model on $\mathbb{Z}^2$, one of the colours only infects a finite number of sites with probability one. We also discuss generalisations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

The chromatic threshold delta(chi) (H, p) of a graph H with respect to the random graph G(n, p) is the infimum over d > 0 such that the following holds with high probability: the family of H-free graphs G subset of G(n, p) with minimum degree delta(G) >= dpn has bounded chromatic number. The study of d. (H) := delta(chi) (H, 1) was initiated in 1973 by Erdos and Simonovits. Recently delta(chi) (H) was determined for all graphs H. It is known that delta(chi) (H, p) =delta(chi) (H) for all fixed p epsilon (0, 1), but that typically delta(chi) (H, p) epsilon not equal delta(chi) (H) if p = 0(1).Here we study the problem for sparse random graphs. We determine delta(chi) (H, p) for most functions p = p(n) when H. {K3, C5}, and also for all graphs H with x(H) is not an element of {3, 4}. (C) 2017 Wiley Periodicals, Inc.

The Ramsey number r ( K 3 , Q n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K N contains either a red n -dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r ( K 3 , Q n )=2 n +1 −1 for every n ∈ℕ, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r ( K 3 , Q n )⩽7000·2 n . Here we show that r ( K 3 , Q n )=(1+ o (1))2 n +1 as n →∞.

We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.

Given an irreducible discrete-time Markov chain on a finite state space, we consider the largest expected hitting time T(α) of a set of stationary measure at least α for α∈(0,1). We obtain tight inequalities among the values of T(α) for different choices of α. One consequence is that T(α) ≤ T(1/2)/α for all α < 1/2. As a corollary we have that, if the chain is lazy in a certain sense as well as reversible, then T(1/2) is equivalent to the chain's mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions T(α) over the domain α∈(0,1/2].

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p c ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

The Ramsey number r(K_s,Q_n) is the smallest positive integer N such that every red-blue colouring of the edges of the complete graph K_N on N vertices contains either a red n-dimensional hypercube, or a blue clique on s vertices. Answering a question of Burr and Erd\H{o}s from 1983, and improving on recent results of Conlon, Fox, Lee and Sudakov, and of the current authors, we show that r(K_s,Q_n) = (s-1) (2^n - 1) + 1 for every s \in \N and every sufficiently large n \in \N.

A connected vertex ordering of a graph G is an ordering v 1 < v 2 < ⋯ < v n of V(G) such that v i has at least one neighbour in {v 1, …, v i − 1}, for every i ∈ {2, …, n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G) = 2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k ) > χ(G k ), for every k ≥ 3. We then prove that for every graph G, χ c (G) ≤ χ(G) + 1. Finally, we prove that deciding if χ c (G) = χ(G), given a graph G, is a NP-hard problem.