A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.

A $(p,q)$ -colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$ -clique. The problem of determining the minimum number of colours, $f(n,p,q)$ , needed to give a $(p,q)$ -colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$ . The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$ , each of which was proved by giving a deterministic construction which combined a $(p,p-1)$ -colouring using few colours with an algebraic colouring.

In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$ -cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$ -colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$ -colourings, which would otherwise make this problem intractable for large values of $p$ . In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$ .

The clustered chromatic number of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$ -colourable with monochromatic components of size at most $c$ . We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.

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

As usual, $P_{n}$ ($n\geq 1$) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$. A graph is called ($P_{5}$, gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$, $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$. We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ($P_{5}$, gem)-free graph $G$.

Is there some absolute $\unicode[STIX]{x1D700}>0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\unicode[STIX]{x1D712}(G^{2})\leqslant (2-\unicode[STIX]{x1D700})\unicode[STIX]{x1D714}(G)^{2}$, where $\unicode[STIX]{x1D714}(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case where $G$ is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.

Let $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

Clark et al. [‘The axiomatizability of topological prevarieties’, Adv. Math.218 (2008), 1604–1653] have shown that, for k≥2, there exists a Boolean topological graph that is k-colourable but not topologically k-colourable; that is, for every ϵ>0, it cannot be coloured by a paintbrush of width ϵ. We generalize this result to show that, for k≥2, there is a Boolean topological graph that is 2-colourable but not topologically k-colourable. This graph is an inverse limit of finite graphs which are shown to exist by an Erdős-style probabilistic argument of Hell and Nešetřil [‘The core of a graph’, Discrete Math.109 (1992), 117–126]. We use the fact that there exists a Boolean topological graph that is 2-colourable but not k-colourable, and some other results (some new and some previously known), to answer the question of which finitely generated topological residual classes of graphs are axiomatizable by universal Horn sentences. A more general version of this question was raised in the above-mentioned paper by Clark et al., and has been investigated by various authors for other structures.

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj, …, Yz) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.