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.

For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa =|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets $A,B$ of $\mathbb{F}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha \gt 0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.

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.

The bipartite independence number of a graph $G$, denoted as $\tilde \alpha (G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $\delta (G)\geq \tilde \alpha (G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta (G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$. In this paper, we show that $\delta (G)\geq \tilde \alpha (G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

We study the noise sensitivity of the minimum spanning tree (MST) of the $n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $\varepsilon \gg n^{-1/3}$, then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if $\varepsilon \ll n^{-1/3}$, the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.

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 $n$.

For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge colouring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge.

Recently, Conlon, Fox, and Wigderson conjectured that for any $0\lt \alpha \lt 1$, the random lower bound $r(B_{\lceil \alpha n\rceil },B_n)\ge (\sqrt{\alpha }+1)^2n+o(n)$ is not tight. In other words, there exists some constant $\beta \gt (\sqrt{\alpha }+1)^2$ such that $r(B_{\lceil \alpha n\rceil },B_n)\ge \beta n$ for all sufficiently large $n$. This conjecture holds for every $\alpha \lt 1/6$ by a result of Nikiforov and Rousseau from 2005, which says that in this range $r(B_{\lceil \alpha n\rceil },B_n)=2n+3$ for all sufficiently large $n$.

We disprove the conjecture of Conlon, Fox, and Wigderson. Indeed, we show that the random lower bound is asymptotically tight for every $1/4\leq \alpha \leq 1$. Moreover, we show that for any $1/6\leq \alpha \le 1/4$ and large $n$, $r(B_{\lceil \alpha n\rceil }, B_n)\le \left (\frac 32+3\alpha \right ) n+o(n)$, where the inequality is asymptotically tight when $\alpha =1/6$ or $1/4$. We also give a lower bound of $r(B_{\lceil \alpha n\rceil }, B_n)$ for $1/6\le \alpha \lt \frac{52-16\sqrt{3}}{121}\approx 0.2007$, showing that the random lower bound is not tight, i.e., the conjecture of Conlon, Fox, and Wigderson holds in this interval.

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.

A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln (n)^{d-j-1}$. This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln (n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.

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.

We show that for every $\eta \gt 0$ every sufficiently large $n$-vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.

Further, we show that in the same setting, $D$ contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$.

Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$-free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

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.

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least

\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}

$\exp\!(n\log n-\Theta (n))$

$\ell$

$(k-\ell )\mid n$

$(k-\ell )\mid k$

$(k-\ell )\nmid k$

$\ell \lt k/2$

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 investigate here the behaviour of a large typical meandric system, proving a central limit theorem for the number of components of a given shape. Our main tool is a theorem of Gao and Wormald that allows us to deduce a central limit theorem from the asymptotics of large moments of our quantities of interest.

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.