In this paper, we study the asymptotic behavior of the generalized Zagreb indices of the classical Erdős–Rényi (ER) random graph G(n, p), as $n\to\infty$. For any integer $k\ge1$, we first give an expression for the kth-order generalized Zagreb index in terms of the number of star graphs of various sizes in any simple graph. The explicit formulas for the first two moments of the generalized Zagreb indices of an ER random graph are then obtained from this expression. Based on the asymptotic normality of the numbers of star graphs of various sizes, several joint limit laws are established for a finite number of generalized Zagreb indices with a phase transition for p in different regimes. Finally, we provide a necessary and sufficient condition for any single generalized Zagreb index of G(n, p) to be asymptotic normal.

We use the framework of multivariate regular variation to analyse the extremal behaviour of preferential attachment models. To this end, we follow a directed linear preferential attachment model for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A Pólya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity.

In 1967, Gerencsér and Gyárfás [16] proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$, there is a monochromatic path on $\lceil (2n+1)/3\rceil $ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erdős [2, 3] regarding the Ramsey numbers of trees (settled in [31]), graphs with bounded maximum degree (settled in [5]), and graphs with bounded degeneracy (settled in [23]).

In 1993, Erdős and Galvin [13] began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest d such that in every $2$-coloring of $K_{\mathbb {N}}$ there is a monochromatic infinite path with upper density at least d? Erdős and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, this problem was finally solved in [7] where it was shown that $d={(12+\sqrt {8})}/{17}$.

This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.

We introduce the notion of echeloned spaces – an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints, the automorphisms of the induced echeloned space are precisely the dilations.

Next, we focus on finite echeloned spaces. They form a Fraïssé class, and we describe its Fraïssé-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this, we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nešetřil and Hubička with a topological-dynamical point of view due to Kechris, Pestov and Todorčević. Finally, using the method of Katětov functors due to Kubiś and Mašulović, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.

In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

We present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large C. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c \gt 0$. We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa’s question is equivalent to our arithmetic Ramsey question.

We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and k-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalisations to arbitrary one-dimensional patterns.

A simple probabilistic argument shows that every r-uniform hypergraph with m edges contains an r-partite subhypergraph with at least $({r!}/{r^r})m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+\Omega(m^{1/2})$, and this is sharp. We prove that if $r\geq 3$, then there is an r-partite subhypergraph with at least $({r!}/{r^r}) m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $({r!}/{r^r}) m+m^{3/4-o(1)}$, which is tight up to the o(1) term. These improve results of Conlon, Fox, Kwan and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming.

A key part of our argument is relating the energy $\mathcal{E}(G)$ of a graph G (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every m edge multigraph G has a cut of size at least $m/2+\Omega({\mathcal{E}(G)}/{\log m})$, which might be of independent interest.

Given a fixed k-uniform hypergraph F, the F-removal lemma states that every hypergraph with few copies of F can be made F-free by the removal of few edges. Unfortunately, for general F, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which F one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all $k\gt2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

A conjecture of Jackson from 1981 states that every d-regular oriented graph on n vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large n. In fact we prove a more general result that for all $\alpha>0$, there exists $n_0=n_0(\alpha )$ such that every d-regular digraph on $n\geq n_0$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and moreover that if G is an oriented graph, then at most $n/(2d+1)$ cycles suffice.

Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\ldots,H_r)$ of graphs if every r-colouring of the edges of G contains a monochromatic copy of $H_i$ in colour i, for some $i \in [\![{r}]\!]$. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Ruciński, predicts the threshold at which the binomial random graph $G_{n,p}$ becomes Ramsey for $(H_1,\ldots,H_r)$ asymptotically almost surely.

In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs $H_1,\ldots,H_r$ by finite families $\mathcal{H}_1,\ldots,\mathcal{H}_r$. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa–Kreuter conjecture.

We study the generalized Ramsey–Turán function $\mathrm {RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an n-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erdős, Sós, Bollobás, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.

A fundamental extremality result due to Sidorenko [‘A partially ordered set of functionals corresponding to graphs’, Discrete Math. 131(1–3) (1994), 263–277] states that among all connected graphs G on k vertices, the k-vertex star maximises the number of graph homomorphisms of G into any graph H. We provide a new short proof of this result using only a simple recursive counting argument for trees and Hölder’s inequality.

We study the length of short cycles on uniformly random metric maps (also known as ribbon graphs) of large genus using a Teichmüller theory approach. We establish that, as the genus tends to infinity, the length spectrum converges to a Poisson point process with an explicit intensity. This result extends the work of Janson and Louf to the multi-faced case.

It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles on $\mathbb {R}^2$ does not.

We show that every $(n,d,\lambda )$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log ^{6}n$ and $\lambda \leq cd$, where $c=\frac {1}{70000}$. This significantly improves a recent result of Glock, Correia, and Sudakov, who obtained a similar result for $d$ that grows polynomially with $n$. The proof is based on a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices, combined with a recent result on connecting designated pairs of vertices by vertex-disjoint paths in $(n,d,\lambda )$-graphs. We believe that the former result is of independent interest and will have further applications.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

This paper is concerned with the growth rate of susceptible–infectious–recovered epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterised by the presence of cliques (fully connected subgraphs). We study epidemics on random intersection graphs with a mixed Poisson degree distribution and show that in the limit of large population sizes the number of infected individuals grows exponentially during the early phase of the epidemic, as is generally the case for epidemics on asymptotically unclustered networks. The Malthusian parameter is shown to satisfy a variant of the classical Euler–Lotka equation. To obtain these results we construct a coupling of the epidemic process and a continuous-time multitype branching process, where the type of an individual is (essentially) given by the length of its infectious period. Asymptotic results are then obtained via an embedded single-type Crump–Mode–Jagers branching process.

A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers $s,t \ge 1$ there exists a finite number $F(s,t)$ such that the vertex set of every digraph of minimum out-degree at least $F(s,t)$ can be partitioned into non-empty parts $A$ and $B$ such that the subdigraphs induced on $A$ and $B$ have minimum out-degree at least $s$ and $t$, respectively.

In this short note, we prove that if $F(2,2)$ exists, then all the numbers $F(s,t)$ with $s,t\ge 1$ exist and satisfy $F(s,t)=\Theta (s+t)$. In consequence, the problem of Alon and Stiebitz reduces to the case $s=t=2$. Moreover, the numbers $F(s,t)$ with $s,t \ge 2$ either all exist and grow linearly, or all of them do not exist.