A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the seminal work of Burr, Erdős, and Lovász to the investigation of other extremal parameters of Ramsey graphs, including the minimum degree.

It is not hard to see that if $G$ is minimally $q$-Ramsey for $H$ we must have $\delta (G) \ge q(\delta (H) - 1) + 1$, and we say that a graph $H$ is $q$-Ramsey simple if this bound can be attained. Grinshpun showed that this is typical of rather sparse graphs, proving that the random graph $G(n,p)$ is almost surely $2$-Ramsey simple when $\frac{\log n}{n} \ll p \ll n^{-2/3}$. In this paper, we explore this question further, asking for which pairs $p = p(n)$ and $q = q(n,p)$ we can expect $G(n,p)$ to be $q$-Ramsey simple.

We first extend Grinshpun’s result by showing that $G(n,p)$ is not just $2$-Ramsey simple, but is in fact $q$-Ramsey simple for any $q = q(n)$, provided $p \ll n^{-1}$ or $\frac{\log n}{n} \ll p \ll n^{-2/3}$. Next, when $p \gg \left ( \frac{\log n}{n} \right )^{1/2}$, we find that $G(n,p)$ is not $q$-Ramsey simple for any $q \ge 2$. Finally, we uncover some interesting behaviour for intermediate edge probabilities. When $n^{-2/3} \ll p \ll n^{-1/2}$, we find that there is some finite threshold $\tilde{q} = \tilde{q}(H)$, depending on the structure of the instance $H \sim G(n,p)$ of the random graph, such that $H$ is $q$-Ramsey simple if and only if $q \le \tilde{q}$. Aside from a couple of logarithmic factors, this resolves the qualitative nature of the Ramsey simplicity of the random graph over the full spectrum of edge probabilities.

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use this to prove a conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006, and a strong version of the Hurewicz theorem in discrete homotopy theory.

In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan’s decomposition theorem for a general class of large-scale interacting systems.

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by Draganić, Glock, and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.

We study the problem of identifying a small number $k\sim n^\theta$, $0\lt \theta \lt 1$, of infected individuals within a large population of size $n$ by testing groups of individuals simultaneously. All tests are conducted concurrently. The goal is to minimise the total number of tests required. In this paper, we make the (realistic) assumption that tests are noisy, that is, that a group that contains an infected individual may return a negative test result or one that does not contain an infected individual may return a positive test result with a certain probability. The noise need not be symmetric. We develop an algorithm called SPARC that correctly identifies the set of infected individuals up to $o(k)$ errors with high probability with the asymptotically minimum number of tests. Additionally, we develop an algorithm called SPEX that exactly identifies the set of infected individuals w.h.p. with a number of tests that match the information-theoretic lower bound for the constant column design, a powerful and well-studied test design.

We investigate the set $T_{n}$ of the possible number of triangles in a graph on n vertices. The first main result says that every natural number less than $\binom {n}{3} - (\sqrt {2}+o(1) )n^{3/2} $ belongs to $T_{n}$. On the other hand, we show that there is a number $m = \binom {n}{3}-( \sqrt {2} +o(1))n^{3/2}$ that is not a member of $T_{n}$. In addition, we prove that there are two interlacing sequences $\binom {n}{3} - ( \sqrt {2} + o(1) )n^{3/2} = c_{1} \le d_{1} \le c_{2} \le d_{2} \le \cdots \le c_{s} \le d_{s} = \binom {n}{3}$ with $| c_{t} - d_{t}| = n-2-\binom {s - t +1}{2}$ such that $( c_{t}, d_{t} ) \cap T_{n} = \emptyset $ for all t. Moreover, we obtain a generalisation of these results for the set of the possible number of copies of a connected graph H in a graph on n vertices.

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

We extend the Kechris–Pestov–Todorčević correspondence to weak Fraïssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders and categories of strong embeddings of trees.

Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$-cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$-cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

The walk matrix associated to an $n\times n$ integer matrix $\mathbf{X}$ and an integer vector $b$ is defined by ${\mathbf{W}} \,:\!=\, (b,{\mathbf{X}} b,\ldots, {\mathbf{X}}^{n-1}b)$. We study limiting laws for the cokernel of $\mathbf{W}$ in the scenario where $\mathbf{X}$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^m$-torsion part of the cokernel, as a group, when $\mathbf{X}$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ${\mathbb{Z}}[x]$-module structure.

The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let ${\mathcal G}(n,W)$ be the random graph obtained by independently including each edge $jk\in \binom{[n]}{2}$ with probability $W_{jk}=W_{kj}$. Given a degree sequence $\textbf{d}=(d_1,\ldots, d_n)$, let ${\mathcal G}(n,\textbf{d})$ denote a uniformly random graph with degree sequence $\textbf{d}$. We couple ${\mathcal G}(n,W)$ and ${\mathcal G}(n,\textbf{d})$ together so that asymptotically almost surely ${\mathcal G}(n,W)$ is a subgraph of ${\mathcal G}(n,\textbf{d})$, where $W$ is some function of $\textbf{d}$. Let $\Delta (\textbf{d})$ denote the maximum degree in $\textbf{d}$. Our coupling result is optimal when $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$, that is, $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for every $i,j\in [n]$. We also have coupling results for $\textbf{d}$ that are not constrained by the condition $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$. For such $\textbf{d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for most pairs $ij\in \binom{[n]}{2}$.

Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$-chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R \subseteq E(G)$ with $|R|\le r$. This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k \ge 4$.

We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$.

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erdős–Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lovász and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part.

In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\ge 4$ and all large $n$, amending the previous conjecture of Pikhurko and Razborov.

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat. 45(1) 289–315.) for non-uniform HSBMs.

Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

We determine the order of the k-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, n-vertex graphs with edge weights $\{a^n_{i,j}\}_{i,j \in [n]}$ that converges to a graphon $W\colon[0,1]^2 \to [0,+\infty)$ in the cut metric. Keeping an edge (i,j) of $G_n$ with probability ${a^n_{i,j}}/{n}$ independently, we obtain a sequence of random graphs $G_n({1}/{n})$. Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of the k-core of random graphs $G_n({1}/{n})$. Our result can also be used to obtain the threshold of appearance of a k-core of order n.

The game of Cops and Robber is traditionally played on a finite graph. The purpose of this article is to introduce and analyze the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer c such that c cops can win the game against one robber, and c only depends on the genus g of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=\Omega (\sqrt {g})$ when the genus g is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.