For a fixed infinite graph H$H$, we study the largest density of a monochromatic subgraph isomorphic to H$H$ that can be found in every two-colouring of the edges of K_{\mathbb{N}}$K_{\mathbb{N}}$. This is called the Ramsey upper density of H$H$ and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs H$H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimisation problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of H$H$, the number of components of H$H$ and the expansion ratio |N(I)|/|I|$|N(I)|/|I|$ of the independent sets of H$H$.

Let G=(V,E)$G=(V,E)$ be a countable graph. The Bunkbed graph of G$G$ is the product graph G \times K_2$G \times K_2$ , which has vertex set V\times \{0,1\}$V\times \{0,1\}$ with “horizontal” edges inherited from G$G$ and additional “vertical” edges connecting (w,0)$(w,0)$ and (w,1)$(w,1)$ for each w \in V$w \in V$ . Kasteleyn’s Bunkbed conjecture states that for each u,v \in V$u,v \in V$ and p\in [0,1]$p\in [0,1]$ , the vertex (u,0)$(u,0)$ is at least as likely to be connected to (v,0)$(v,0)$ as to (v,1)$(v,1)$ under Bernoulli- p$p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the p \uparrow 1$p \uparrow 1$ limit in the sense that for each finite graph G$G$ there exists \varepsilon (G)\gt 0$\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for p \geqslant 1-\varepsilon (G)$p \geqslant 1-\varepsilon (G)$ .

Let {\mathbb{G}(n_1,n_2,m)}${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m-edge subgraph of the complete bipartite graph {K_{n_1,n_2}}${K_{n_1,n_2}}$ with bipartition (V_1, V_2)$(V_1, V_2)$ , where n_i = |V_i|$n_i = |V_i|$ , i=1,2$i=1,2$ . Given a real number p \in [0,1]$p \in [0,1]$ such that d_1 \,{:\!=}\, pn_2$d_1 \,{:\!=}\, pn_2$ and d_2 \,{:\!=}\, pn_1$d_2 \,{:\!=}\, pn_1$ are integers, let \mathbb{R}(n_1,n_2,p)$\mathbb{R}(n_1,n_2,p)$ be a random subgraph of {K_{n_1,n_2}}${K_{n_1,n_2}}$ with every vertex v \in V_i$v \in V_i$ of degree d_i$d_i$ , i = 1, 2$i = 1, 2$ . In this paper we determine sufficient conditions on n_1,n_2,p$n_1,n_2,p$ and m under which one can embed {\mathbb{G}(n_1,n_2,m)}${\mathbb{G}(n_1,n_2,m)}$ into \mathbb{R}(n_1,n_2,p)$\mathbb{R}(n_1,n_2,p)$ and vice versa with probability tending to 1. In particular, in the balanced case n_1=n_2$n_1=n_2$ , we show that if p\gg\log n/n$p\gg\log n/n$ and 1 - p \gg \left(\log n/n \right)^{1/4}$1 - p \gg \left(\log n/n \right)^{1/4}$ , then for some m\sim pn^2$m\sim pn^2$ , asymptotically almost surely one can embed {\mathbb{G}(n_1,n_2,m)}${\mathbb{G}(n_1,n_2,m)}$ into \mathbb{R}(n_1,n_2,p)$\mathbb{R}(n_1,n_2,p)$ , while for p\gg\left(\log^{3} n/n\right)^{1/4}$p\gg\left(\log^{3} n/n\right)^{1/4}$ and 1-p\gg\log n/n$1-p\gg\log n/n$ the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than (n \log n)^{3/4}$(n \log n)^{3/4}$ .

We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.

We find an asymptotic enumeration formula for the number of simple r$r$ -uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random r$r$ -uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random r$r$ -uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.

Given a family \mathcal{F}$\mathcal{F}$ of bipartite graphs, the Zarankiewicz number z(m,n,\mathcal{F})$z(m,n,\mathcal{F})$ is the maximum number of edges in an m$m$ by n$n$ bipartite graph G$G$ that does not contain any member of \mathcal{F}$\mathcal{F}$ as a subgraph (such G$G$ is called \mathcal{F}$\mathcal{F}$ -free). For 1\leq \beta \lt \alpha \lt 2$1\leq \beta \lt \alpha \lt 2$ , a family \mathcal{F}$\mathcal{F}$ of bipartite graphs is (\alpha,\beta )$(\alpha,\beta )$ -smooth if for some \rho \gt 0$\rho \gt 0$ and every m\leq n$m\leq n$ , z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$ . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any (\alpha,\beta )$(\alpha,\beta )$ -smooth family \mathcal{F}$\mathcal{F}$ , there exists k_0$k_0$ such that for all odd k\geq k_0$k\geq k_0$ and sufficiently large n$n$ , any n$n$ -vertex \mathcal{F}\cup \{C_k\}$\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least \rho (\frac{2n}{5}+o(n))^{\alpha -1}$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real \delta \gt 0$\delta \gt 0$ , there exists k_0$k_0$ such that for all odd k\geq k_0$k\geq k_0$ and sufficiently large n$n$ , any n$n$ -vertex \mathcal{F}\cup \{C_k\}$\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least \delta n^{\alpha -1}$\delta n^{\alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families \mathcal{F}$\mathcal{F}$ consisting of the single graph K_{s,t}$K_{s,t}$ when t\gg s$t\gg s$ . We also prove an analogous result for C_{2\ell }$C_{2\ell }$ -free graphs for every \ell \geq 2$\ell \geq 2$ , which complements a result of Keevash, Sudakov and Verstraëte.

A chordal graph is a graph with no induced cycles of length at least 4$4$. Let f(n,m)$f(n,m)$ be the maximal integer such that every graph with n$n$ vertices and m$m$ edges has a chordal subgraph with at least f(n,m)$f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating f(n,m)$f(n,m)$. In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of f(n,n^2/4+1)$f(n,n^2/4+1)$ and made a conjecture on the value of f(n,n^2/3+1)$f(n,n^2/3+1)$. In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining f(n,m)$f(n,m)$ asymptotically for all m$m$ and exactly for m \leq n^2/3+1$m \leq n^2/3+1$.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

For a random binary noncoalescing feedback shift register of width n$n$ , with all 2^{2^{n-1}}$2^{2^{n-1}}$ possible feedback functions f$f$ equally likely, the process of long cycle lengths, scaled by dividing by N=2^n$N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in \mathcal{S}_N$\mathcal{S}_N$ , with all N!$N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.

In this paper, we study asymmetric Ramsey properties of the random graph G_{n,p}$G_{n,p}$ . Let r \in \mathbb{N}$r \in \mathbb{N}$ and H_1, \ldots, H_r$H_1, \ldots, H_r$ be graphs. We write G_{n,p} \to (H_1, \ldots, H_r)$G_{n,p} \to (H_1, \ldots, H_r)$ to denote the property that whenever we colour the edges of G_{n,p}$G_{n,p}$ with colours from the set [r] \,{:\!=}\, \{1, \ldots, r\}$[r] \,{:\!=}\, \{1, \ldots, r\}$ there exists i \in [r]$i \in [r]$ and a copy of H_i$H_i$ in G_{n,p}$G_{n,p}$ monochromatic in colour i$i$ . There has been much interest in determining the asymptotic threshold function for this property. In several papers, Rödl and Ruciński determined a threshold function for the general symmetric case; that is, when H_1 = \cdots = H_r$H_1 = \cdots = H_r$ . A conjecture of Kohayakawa and Kreuter from 1997, if true, would fully resolve the asymmetric problem. Recently, the 1$1$ -statement of this conjecture was confirmed by Mousset, Nenadov and Samotij.

Building on work of Marciniszyn, Skokan, Spöhel and Steger from 2009, we reduce the 0$0$ -statement of Kohayakawa and Kreuter’s conjecture to a certain deterministic subproblem. To demonstrate the potential of this approach, we show this subproblem can be resolved for almost all pairs of regular graphs. This therefore resolves the 0$0$ -statement for all such pairs of graphs.

A conjecture of Alon, Krivelevich and Sudakov states that, for any graph F$F$ , there is a constant c_F \gt 0$c_F \gt 0$ such that if G$G$ is an F$F$ -free graph of maximum degree \Delta$\Delta$ , then \chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs F$F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if G$G$ is K_{t,t}$K_{t,t}$ -free, then \chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$ as \Delta \to \infty$\Delta \to \infty$ . We improve this bound to (1+o(1)) \Delta/\log\!\Delta$(1+o(1)) \Delta/\log\!\Delta$ , making the constant factor independent of t$t$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.

We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group G$G$ . We introduce the switched random walk determined by a finite set of probability distributions on G$G$ , prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any d$d$ -regular graph on n$n$ vertices contains a spanning subgraph in which the number of vertices of each degree between 0$0$ and d$d$ deviates from \frac{n}{d+1}$\frac{n}{d+1}$ by at most 2$2$ . The second is that every graph on n$n$ vertices with minimum degree \delta$\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed \frac{n}{\delta +1}+2$\frac{n}{\delta +1}+2$ . Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices n$n$ . In particular we show that if d^3 \log n \leq o(n)$d^3 \log n \leq o(n)$ then every d$d$ -regular graph with n$n$ vertices contains a spanning subgraph in which the number of vertices of each degree between 0$0$ and d$d$ is (1+o(1))\frac{n}{d+1}$(1+o(1))\frac{n}{d+1}$ . We also prove that any graph with n$n$ vertices and minimum degree \delta$\delta$ contains a spanning subgraph in which no degree is repeated more than (1+o(1))\frac{n}{\delta +1}+2$(1+o(1))\frac{n}{\delta +1}+2$ times.

We show that the size-Ramsey number of the \sqrt{n} \times \sqrt{n}$\sqrt{n} \times \sqrt{n}$ grid graph is O(n^{5/4})$O(n^{5/4})$, improving a previous bound of n^{3/2 + o(1)}$n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.

We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class \mathcal H$\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in \mathcal H$\mathcal H$. Our main result is that there exists a geometric graph with n$n$ vertices and O\!\left(n \log n\right)$O\!\left(n \log n\right)$ edges that is universal for n$n$-vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with n$n$ vertices and O\!\left(n \log n\right)$O\!\left(n \log n\right)$ edges that contains every n$n$-vertex forest as a subgraph. The upper bound of O\!\left(n \log n\right)$O\!\left(n \log n\right)$ edges cannot be improved, even if more than n$n$ vertices are allowed. We also prove that every n$n$-vertex convex geometric graph that is universal for n$n$-vertex outerplanar graphs has a near-quadratic number of edges, namely \Omega _h(n^{2-1/h})$\Omega _h(n^{2-1/h})$, for every positive integer h$h$; this almost matches the trivial O(n^2)$O(n^2)$ upper bound given by the n$n$-vertex complete convex geometric graph. Finally, we prove that there exists an n$n$-vertex convex geometric graph with n$n$ vertices and O\!\left(n \log n\right)$O\!\left(n \log n\right)$ edges that is universal for n$n$-vertex caterpillars.

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running k$k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when k$k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of \Omega ((n/k) \log n)$\Omega ((n/k) \log n)$ on the stationary cover time, holding for any n$n$ -vertex graph G$G$ and any 1 \leq k =o(n\log n )$1 \leq k =o(n\log n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

We study several parameters of a random Bienaymé–Galton–Watson tree T_n$T_n$ of size n$n$ defined in terms of an offspring distribution \xi$\xi$ with mean 1$1$ and nonzero finite variance \sigma ^2$\sigma ^2$ . Let f(s)=\mathbb{E}\{s^\xi \}$f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable \xi$\xi$ . We show that the independence number is in probability asymptotic to qn$qn$ , where q$q$ is the unique solution to q = f(1-q)$q = f(1-q)$ . One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to \log n/\log (1/f'(1-q))$\log n/\log (1/f'(1-q))$ . Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If p_1 = \mathbb{P}\{\xi =1\}\gt 0$p_1 = \mathbb{P}\{\xi =1\}\gt 0$ , then we show that the maximum leaf-height over all nodes in T_n$T_n$ is in probability asymptotic to \log n/\log (1/p_1)$\log n/\log (1/p_1)$ . If p_1 = 0$p_1 = 0$ and $\kappa$ is the first integer $i\gt 1$ with $\mathbb{P}\{\xi =i\}\gt 0$ , then the leaf-height is in probability asymptotic to $\log _\kappa \log n$ .

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ . We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$ , where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.

Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.

We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$ -universality.

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.

In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$-uniform hypergraphs by proving a $3$-uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.