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.

Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$ -blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$ -blanket times of the random walks if the $\varepsilon$ -blanket time of the limiting diffusion is continuous at $\varepsilon$ with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the $\varepsilon$ -blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$.

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.

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 $c$ .

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$ , and let $\phi \,:\, \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as

\begin{equation*} {\sf avgStretch}(\phi ) = {\mathbb E}[{{\sf dist}}(\phi (x),\phi (x'))], \end{equation*}

$x,x' \in \{0,1\}^{n-1}$

In this paper, we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results.

• For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi _A \,:\, \{0,1\}^{n-1} \to A$ such that ${\sf avgStretch}(\phi _A) = O\left(\sqrt{n}\right)$ .

• For $n = 3^k$ let ${A_{\textsf{rec-maj}}} = \{x \in \{0,1\}^n \,:\,{\textsf{rec-maj}}(x) = 1\}$ , where ${\textsf{rec-maj}} \,:\, \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3’s. There exists a bijection $\phi _{{\textsf{rec-maj}}} \,:\, \{0,1\}^{n-1} \to{A_{\textsf{rec-maj}}}$ such that ${\sf avgStretch}(\phi _{{\textsf{rec-maj}}}) = O(1)$ .

• Let ${A_{{\sf tribes}}} = \{x \in \{0,1\}^n \,:\,{\sf tribes}(x) = 1\}$ . There exists a bijection $\phi _{{\sf tribes}} \,:\, \{0,1\}^{n-1} \to{A_{{\sf tribes}}}$ such that ${\sf avgStretch}(\phi _{{\sf tribes}}) = O(\!\log (n))$ .

These results answer the questions raised by Benjamini, Cohen, and Shinkar (Isr. J. Math 2016).

We show that the $4$ -state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$ -regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$ . This is tight since it is known that there are multiple Gibbs measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and $d\geq 4$ . We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$ -state Potts model on the $(d+1)$ -regular tree for $w\geq 1-\dfrac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$ .

In this note, we give a precise description of the limiting empirical spectral distribution for the non-backtracking matrices for an Erdős-Rényi graph $G(n,p)$ assuming $np/\log n$ tends to infinity. We show that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably, and then, we use Tao and Vu’s replacement principle and the Bauer-Fike theorem to show that the partly derandomized spectrum is, in fact, very close to the original spectrum.

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 extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$-cycles.