The slice decomposition is a bijective method for enumerating planar maps (graphs embedded in the sphere) with control over face degrees. In this paper, we extend the slice decomposition to the richer setting of hypermaps, naturally interpreted as properly face-bicolored maps, where the degrees of faces of each color can be controlled separately. This setting is closely related with the two-matrix model and the Ising model on random maps, which have been intensively studied in theoretical physics, leading to several enumerative formulas for hypermaps that were still awaiting bijective proofs.

Generally speaking, the slice decomposition consists in cutting along geodesics. A key feature of hypermaps is that the geodesics along which we cut are directed, following the canonical orientation of edges imposed by the coloring. This orientation requires us to introduce an adapted notion of slices, which admit a recursive decomposition that we describe.

Using these slices as fundamental building blocks, we obtain new bijective decompositions of several families of hypermaps: disks (pointed or not) with a monochromatic boundary, cylinders with monochromatic boundaries (starting with trumpets or cornets having one geodesic boundary), and disks with a “Dobrushin” boundary condition. In each case, the decomposition ultimately expresses these objects as sequences of slices whose increments correspond to downward skip-free (Łukasiewicz-type) walks subject to natural constraints.

Our approach yields bijective proofs of several explicit expressions for hypermap generating functions. In particular, we provide a combinatorial explanation of the algebraicity and of the existence of rational parametrizations for these generating functions when face degrees are bounded.

A set X is called Euclidean Ramsey if, for any k and sufficiently large n, every k-colouring of $\mathbb {R}^n$ contains a monochromatic congruent copy of X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus. They asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. It is not too difficult to show that if a set is not spherical then it is not Euclidean Ramsey either, but the converse is very much open despite extensive research over the years.

On the other hand, the block sets conjecture is a purely combinatorial, Hales-Jewett type of statement, concerning ‘blocks in large products’, introduced by Leader, Russell and Walters. If true, the block sets conjecture would imply that every transitive set (a set whose symmetry group acts transitively) is Euclidean Ramsey. As for the question above, the block sets conjecture remains very elusive, being known only in a few cases.

In this paper we show that the sizes of the blocks in the block sets conjecture cannot be bounded, even for templates over the alphabet of size 3. We also show that for the first nontrivial template, namely $123$, the blocks may be taken to be of size $2$ (for any number of colours). This is best possible; all previous bounds were ‘tower-type’ large.

We derive exact formulas for the proportions of derangements and of derangements of p-power order in the affine classical groups $\operatorname {\mathrm {AU}}_m(q)$, $\operatorname {\mathrm {ASp}}_{2m}(q)$, $\operatorname {\mathrm {AO}}_{2m+1}(q)$ and $\operatorname {\mathrm {AO}}^{\pm }_{2m}(q)$, where p denotes the characteristic of the defining finite field.

In the unitary case, the proofs of the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda =(\lambda _1, \dots , \lambda _m)$ into m parts, with $\lambda _1\ge \dots \ge \lambda _m$, such that either $\lambda _1=1$ or $\lambda _{k-1}>\lambda _k=k$ for some $k \in \{2, \dots ,m\}$.

In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three q-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

Consider n points independently sampled from a density p of class $\mathcal{C}^2$ on a smooth compact d-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the random walk visiting these points according to a transition kernel K. We study the almost sure uniform convergence of the generator of this process to the diffusive Laplace–Beltrami operator when n tends to infinity, from which we establish the convergence of the random walk to a diffusion process on the manifold. In contrast to known results, our result does not require the kernel K to be continuous, which covers the cases of walks exploring k-nearest neighbor (kNN) and geometric graphs, and convergence rates are given. The distance between the random walk generator and the limiting operator is separated into several terms: a statistical term, related to the law of large numbers, is treated with concentration tools and an approximation term that we control with tools from differential geometry. The case of kNN Laplacians is detailed. The convergence of the stochastic processes having these operators as generators is also studied, by establishing additional tightness results of their distributions on the space of càdlàg functions.

We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.

We introduce a quantum automorphism group for hypergraphs, which turns out to generalize the quantum automorphism group of Bichon for classical graphs. Further, we show that our quantum automorphism group acts on hypergraph $C^*$-algebras as recently defined. In particular, this action generalizes the one on graph $C^*$-algebras by Schmidt–Weber in 2018.

End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology.

Our main result is that every end-space can be topologically represented by a special order tree. Our main proof ingredient is a structure theorem that we introduce, which carves out the order-tree-like structure of any graph in such a way that there is a natural bijection between the ends of the graph and the limit-type down-closed chains of the order-tree.

Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{def}{=} \{1, \dots, n\}$ with distance set D. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B \in \mathcal{F}$. Kleitman’s celebrated discrete isodiametric inequality states that $f_D(n)$ is maximised at Hamming balls of radius $d/2$ when $D = \{1, \dots, d\}$. We study the generalisation where D is a set of arithmetic progression and determine $f_D(n)$ asymptotically for all homogeneous D. In the special case when D is an interval, our result confirms a conjecture of Huang, Klurman and Pohoata. Moreover, we demonstrate a dichotomy in the growth of $f_D(n)$, showing linear growth in n when D is a non-homogeneous arithmetic progression. Different from previous combinatorial and spectral approaches, we deduce our results by converting the restricted distance problems to restricted intersection problems. Our proof ideas can be adapted to prove upper bounds on t-distance sets in Hamming cubes (also known as binary t-codes), which has been extensively studied by algebraic combinatorialists community, improving previous bounds from polynomial methods and optimisation approaches.

We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon, Fox, Gunby, He, Mubayi, Suk, and Verstraëte disproved the folklore conjecture that $r(H, K_n^{(3)})$ always grows polynomially in $n$. In this paper, we show that much larger growth rates are possible in higher uniformity. In uniformity $k\ge 4$, we prove that for any constant $C\gt 0$, there exists a linear $k$-uniform hypergraph $H$ for which

\begin{equation*} r(H,K_n^{(k)}) \geq {\textrm {twr}}_{k-2}(2^{(\log n)^C}). \end{equation*}

Given a permutation group G, the derangement graph of G is defined with vertex set G, where two elements x and y are adjacent if and only if $xy^{-1}$ is a derangement. We establish that if G is transitive with degree exceeding 30, then the derangement graph of G contains a complete subgraph with four vertices. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that $|A : G| = 3$ and U is a subgroup of G satisfying $G = \bigcup _{a \in A} U^a$, then $|G : U| \leq 10$. This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.

We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincaré polynomial, (2) characterize when these varieties are irreducible, and (3) show that all are reduced schemes. We prove that the singular locus of any nilpotent codimension one Hessenberg variety is also a Hessenberg variety. A key tool in our analysis is a new result applying to all (type A) Hessenberg varieties without any restriction on codimension, which states that their Poincaré polynomials can be computed by counting the points in the corresponding variety defined over a finite field. The results below were motivated by earlier work of the authors studying the precise relationship between Hessenberg and Schubert varieties, and we obtain a corollary extending the results from that paper to all codimension one (type A) Schubert varieties.

We study a family of Crump–Mode–Jagers branching processes in a random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of Iyer and the author (‘On the structure of genealogical trees associated with explosive Crump–Mode–Jagers branching processes’, arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of Iyer and the author and study a large range of previously unattainable cases.

We study the number of triangles $T_n$ in the sparse $\beta$-model on n vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin–Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between the normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$ as $n\to\infty$.

Let $X_1,\ldots, X_n$ be independent integers distributed uniformly on [M], $M\ge 2$. A partition S of [n] into $\nu$ non-empty subsets $S_1,\ldots, S_{\nu}$ is called perfect if all $\nu$ values $\sum_{j\in S_{\alpha}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$. In 2001, for $\nu=2$, Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa\;:\!=\; \lim {{n}/{\log M}}>{{1}/{\log 2}}$, and that with high probability no perfect partition exists if $\kappa<{{1}/{\log 2}}$. Responding to a question by George Varghese, we prove that for $\nu\ge 3$ with high probability no perfect partition exists if $\kappa<{{2}/{\log \nu}}$, which is twice as large as the naive threshold $1/\log 3$ for $\nu=3$. We identify the range of $\kappa$ where the expected number of perfect partitions is exponentially high. We show that for $\kappa> {{2(\nu-1)}/{\log[(1-2\nu^{-2})^{-1}]}}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+\nu^2)^{-1}$, i.e. below $1/\nu$, the limiting probability that $\sum_j X_j$ is divisible by $\nu$.

An orthomorphism of a finite group G is a bijection $\phi \colon G\to G$ such that $g\mapsto g^{-1}\phi (g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when G is abelian, for any $k\geq 2$ dividing $|G|-1$, there exists an orthomorphism of G fixing the identity and permuting the remaining elements as products of disjoint k-cycles as long as the Sylow $2$-subgroups of G are trivial or noncyclic. We prove this conjecture for all sufficiently large groups.

Bender, Coley, Robbins, and Rumsey posed the problem of counting the number of subspaces which have a given profile with respect to a linear endomorphism defined on a finite vector space. We settle this problem in full generality by giving an explicit counting formula involving symmetric functions. This formula can be expressed compactly in terms of a Hall scalar product involving dual q-Whittaker functions and another symmetric function that is determined by conjugacy class invariants of the endomorphism. As corollaries, we obtain new combinatorial interpretations for the coefficients in the q-Whittaker expansions of several symmetric functions. These include the power sum, complete homogeneous, products of modified Hall–Littlewood functions, and certain products of q-Whittaker functions. These results are used to derive a formula for the number of anti-invariant subspaces (as defined by Barría and Halmos) with respect to an arbitrary operator. We also give an application to an open problem in Krylov subspace theory.

We introduce a module-theoretic approach and a linear-programming method to compute the matricial dimensions of numerical semigroups. We compute the matricial dimension of every numerical semigroup with Frobenius number at most $10$ or genus at most $6$. Many of these evaluations are beyond the scope of previous techniques.

Given a collection $\mathcal{D} =\{D_1,D_2,\ldots ,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is transversal in $\mathcal{D}$ if there exists a bijection $\varphi \,:\,E(H)\rightarrow [m]$ such that $e \in E(D_{\varphi (e)})$ for all $e\in E(H)$. Ghouila-Houri proved that any $n$-vertex digraph with minimum semi-degree at least $\frac {n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalisation of Ghouila-Houri’s theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee, and Seo. Our proof utilises the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac’s theorem, which was proved by Joos and Kim.

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.