We show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$, there exists $\alpha \gt 0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$. This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval $[0,\lambda ]$. As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all $\lambda \lt 1/(e^{B + 1} \hat C_\phi )$ where $\hat C_\phi$ is a temperedness parameter and $B$ is the stability constant of $\phi$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $\lambda \lt e/\Delta _\phi$, where $\Delta _\phi$ is the potential-weighted connective constant of the potential $\phi$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

Given a graph $H$, let us denote by $f_\chi (H)$ and $f_\ell (H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$. A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$, for which it is known that $2t-o(t) \le f_\ell (K_t) \le O(t (\!\log \log t)^6)$. Thus, $f_\ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi (K_t)$ by at least a constant factor. The so-called $H$-Hadwiger’s conjecture, proposed by Seymour, asks to prove that $f_\chi (H)={\textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger’s conjecture).

In this paper, we prove several new lower bounds on $f_\ell (H)$, thus exploring the limits of a list colouring extension of $H$-Hadwiger’s conjecture. Our main results are:

• For every $\varepsilon \gt 0$ and all sufficiently large graphs $H$ we have $f_\ell (H)\ge (1-\varepsilon )({\textrm{v}}(H)+\kappa (H))$, where $\kappa (H)$ denotes the vertex-connectivity of $H$.

• For every $\varepsilon \gt 0$ there exists $C=C(\varepsilon )\gt 0$ such that asymptotically almost every $n$-vertex graph $H$ with $\left \lceil C n\log n\right \rceil$ edges satisfies $f_\ell (H)\ge (2-\varepsilon )n$.

The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$-minor-free graphs is separated from the desired value of $({\textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_\ell (H)$ is separated from $({\textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$. Conceptually these results indicate that the graphs $H$ for which $f_\ell (H)$ is close to the conjectured value $({\textrm{v}}(H)-1)$ for $f_\chi (H)$ are typically rather sparse.

Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For $k$ colours and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-\varepsilon }$. Following a sequence of gradual improvements for $k\geq n$ colours, the central open question is to resolve the gap between $\Omega (n)$ and $\mathcal{O}(n\log \log n)$ for $k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$.

We study generalised quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)^q+q$ vertices; subsequently, Lovász refined the argument to show that graphs with $4(2q+3)^8$ vertices suffice. Our results imply that the structure of generalised quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q^2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.

The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $\epsilon \gt 0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl ((2019) Comb. Probab. Comput. 28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.

Given a fixed graph $H$ and a constant $c \in [0,1]$, we can ask what graphs $G$ with edge density $c$ asymptotically maximise the homomorphism density of $H$ in $G$. For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximising $G$ is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs $H$ and densities $c$ such that the optimising graph $G$ is neither the quasi-star nor the quasi-clique (Day and Sarkar, SIAM J. Discrete Math. 35(1), 294–306, 2021). We also show that for $c$ large enough all graphs $H$ maximise on the quasi-clique (Gerbner et al., J. Graph Theory 96(1), 34–43, 2021), and for any $c \in [0,1]$ the density of $K_{1,2}$ is always maximised on either the quasi-star or the quasi-clique (Ahlswede and Katona, Acta Math. Hung. 32(1–2), 97–120, 1978). Finally, we extend our results to uniform hypergraphs.

In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $\alpha n$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$. Our proofs make use of a variant of an absorbing method of Montgomery.

We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$-th power of their Euclidean distance, with $p\gt 0$. In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$, we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

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.

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then

\begin{equation*} H(X_1+\cdots +X_{n+1}) \geq H(X_1+\cdots +X_{n}) + \frac {1}{2}\log {\Bigl (\frac {n+1}{n}\Bigr )} - o(1) \end{equation*}

$H(X_1) \to \infty$

$H(X_1)$

$U_1,\ldots,U_n$

$(0,1)$

\begin{equation*} h(X_1+\cdots +X_n + U_1+\cdots +U_n) = H(X_1+\cdots +X_n) + o(1), \end{equation*}

$H(X_1) \to \infty$

$h$

$o(1)$

We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta )$). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a much larger zero-free disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$. We establish this in the case of linear hypertrees.

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$); and the class of $K_t$-minor-free graphs has underlying treewidth $t-2$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n \gt 0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int _0^1 b_n(x) dx = 0$. We offer a related characterization: $b_1(x) = x - 1/2$ and $({-}1)^{n-1} b_n(x)$ for $n \gt 0$ is the $n$-fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta$(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$. This result has a novel combinatorial analog, the Bernoulli clock: mark the hours of a $2 n$ hour clock by a uniformly random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$, meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$, then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$, and so on. Starting from hour $0 = 2n$, move clockwise to the first hour marked $1$, continue clockwise to the first hour marked $2$, and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$. We show that for each positive integer $n$, the event $( I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$, where $n! b_n(0) = B_n(0)$ is the $n$th Bernoulli number. For $ 1 \le k \le 2 n$, the difference $\delta _n(k)\,:\!=\, 1/(2n) -{\mathbb{P}}( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $\delta _n( 2 n + 1 - k) = ({-}1)^n \delta _n(k)$, which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials $b_n(1-x) = ({-}1)^n b_n(x)$.

We prove that for every tree $T$ of radius $h$, there is an integer $c$ such that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that radius is the right parameter to consider in this setting, and $2h-1$ is the best possible bound.

We consider bond percolation on high-dimensional product graphs $G=\square _{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$, where $d$ is the average degree of the host graph.

In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651–670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

Let $\mathcal{F}$ be an intersecting family. A $(k-1)$-set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$. Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$. In the present paper, we show that for $n\geq 28k$, $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.

For a graph $H$ and a hypercube $Q_n$, $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$, $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$ has a positive or zero Turán density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of graphs, ones having so-called partite representation, that have zero Turán density. He asked whether this gives a characterisation, that is, whether a graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of graphs which have no partite representation, but on the other hand, have zero Turán density. In addition, we show that any graph whose every block has partite representation has zero Turán density in a hypercube.