In this paper we study a variation of the random $k$-SAT problem, called polarised random $k$-SAT, which contains both the classical random $k$-SAT model and the random version of monotone $k$-SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter $p$, and in half of the clauses each variable occurs negated with probability $p$ and pure otherwise, while in the other half the probabilities are interchanged. For $p=1/2$ we get the classical random $k$-SAT model, and at the other extreme we have the fully polarised model where $p=0$, or 1. Here there are only two types of clauses: clauses where all $k$ variables occur pure, and clauses where all $k$ variables occur negated. That is, for $p=0$, and $p=1$, we get an instance of random monotone $k$-SAT.

We show that the threshold of satisfiability does not decrease as $p$ moves away from $\frac{1}{2}$ and thus that the satisfiability threshold for polarised random $k$-SAT with $p\neq \frac{1}{2}$ is an upper bound on the threshold for random $k$-SAT. Hence the satisfiability threshold for random monotone $k$-SAT is at least as large as for random $k$-SAT, and we conjecture that asymptotically, for a fixed $k$, the two thresholds coincide.

Given $\alpha \gt 0$ and an integer $\ell \geq 5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell ^{-}$, a tight cycle on $\ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.

We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$, while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$, and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.

In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$-uniform hypergraphs by proving a $3$-uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$-vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that contains every $n$-vertex forest as a subgraph. The upper bound of $O\!\left(n \log n\right)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega _h(n^{2-1/h})$, for every positive integer $h$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$-vertex caterpillars.

A chordal graph is a graph with no induced cycles of length at least $4$. Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$. In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$. In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$.

The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is

• at least $5^{n/2}$ for $n$ even, $a=b=n$ ,

• equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr )\bigl (\lceil \frac{n}{2}\rceil +1\bigr )$ if $a=2$ and $b=n\ge 4$ ,

• at most $|\cup _{i=1}^m A_i|$ ,

• asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$ ),

• asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

A random two-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log (n)$ . While there are many families of graphs whose expected number of faces is $\Theta (n)$ , none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any $n$ -vertex multigraph, the expected number of faces in a random two-cell embedding is at most $2n\log (2\mu )$ , where $\mu$ is the maximum edge-multiplicity. This bound is best possible up to a constant factor.

For a $k$ -uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega (1/n^{k - 1})$ , ${\mathrm{coker}}(M(\mathcal{H}))$ is torsion-free.

Given partially ordered sets (posets) $(P, \leq _P\!)$ and $(P^{\prime}, \leq _{P^{\prime}}\!)$ , we say that $P^{\prime}$ contains a copy of $P$ if for some injective function $f\,:\, P\rightarrow P^{\prime}$ and for any $X, Y\in P$ , $X\leq _P Y$ if and only if $f(X)\leq _{P^{\prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$ -dimensional Boolean lattice are coloured in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$ -dimensional Boolean lattice $Q_n$ , as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$ , that is a poset on elements $A, B, C$ such that $B\gt C$ , $A\gt C$ , and $A$ and $B$ incomparable, or a poset $\Lambda$ , its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $\Lambda$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$ . Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$ . This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + \Theta \left(\frac{n}{\log n}\right)$ .

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega ((n/k) \log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 \leq k =o(n\log n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $\mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.

The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ .

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$ -vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ . We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$ , then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ .

We consider supercritical site percolation on the $d$ -dimensional hypercube $Q^d$ . We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$ . This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.

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 $f(n,8,8)$ .

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

Motivated by problems from compressed sensing, we determine the threshold behaviour of a random $n\times d \pm 1$ matrix $M_{n,d}$ with respect to the property ‘every $s$ columns are linearly independent’. In particular, we show that for every $0\lt \delta \lt 1$ and $s=(1-\delta )n$ , if $d\leq n^{1+1/2(1-\delta )-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-\delta )+o(1)}$ then with high probability there are some $s$ linearly dependent columns.

Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput. 17 815–821), Kang and Oum (2019, Combin. Probab. Comput. 28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc. 98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.