We introduce and study different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

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.

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to apply to hypergraphs in both round-based and lazy settings. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads – if some fixed proportion of vertices are on fire in a hyperedge, then in the next round, the entire hyperedge catches fire.

We obtain bounds on the burning numbers of general hypergraphs, and introduce the concept of the burning distribution, which describes how the burning numbers change as the proportion parameter ranges over $(0,1)$. We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

Free Steiner triple systems (STSs) are infinite structures that are naturally characterised by a universal property. We consider the class of free STSs from a model-theoretic viewpoint. We show that free STSs on any number of generators are elementarily equivalent. We axiomatise their theory and show that it is stable.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial t-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte T-designs. We explore when a T-design is also a $T'$-design, where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

In 1967, Klarner proposed a problem concerning the existence of reflecting n-queens configurations. The problem considers the feasibility of placing n mutually nonattacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ “reflecting strip” of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which n is it possible to pair up the integers 1 through n with the integers $n+1$ through $2n$ such that no two of the sums or differences of the n pairs of integers are the same. We prove the existence of reflecting n-queens configurations for all sufficiently large n, thereby resolving both Slater’s and Klarner’s questions for all but a finite number of integers.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

We generalize Baker–Bowler’s theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets and orthogonal vector sets, and establish basic properties on functoriality, duality and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular if and only if it is representable over ${\mathbb F}_2$ and ${\mathbb F}_3$, which was originally shown by Geelen [16], and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over ${\mathbb F}_3$ and ${\mathbb F}_4$.

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

For every positive integer d, we show that there must exist an absolute constant $c \gt 0$ such that the following holds: for any integer $n \geqslant cd^{7}$ and any red-blue colouring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$, there must exist either a d-dimensional subspace for which all of its one-dimensional subspaces get coloured red or a 2-dimensional subspace for which all of its one-dimensional subspaces get coloured blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid N, the class of N-free, claw-free binary matroids is polynomially $\chi$-bounded.

Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A \subset \mathbb{F}_{2}^{n}$ with density $\alpha \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\alpha$ is large with respect to n, which utilises ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.

We give an explicit quadratic Gröbner basis for generalized Chow rings of supersolvable built lattices, with the help of the operadic structure on geometric lattices introduced in a previous article. This shows that the generalized Chow rings associated to minimal building sets of supersolvable lattices are Koszul. As another consequence, we get that the cohomology algebras of the components of the extended modular operad in genus $0$ are Koszul.

We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this to categories with finite products, and then further to internal versions for certain Cartesian closed monoidal categories with pullbacks.

Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, each isomorphic to H, that cover the vertices of G. If G is the complete graph $K_n$ equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for $H=K_2$, for example. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n = \Theta(n^{1-1/d^*})$ (where $d^*$ is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings.

An identity that is reminiscent of the Littlewood identity plays a fundamental role in recent proofs of the facts that alternating sign triangles are equinumerous with totally symmetric self-complementary plane partitions and that alternating sign trapezoids are equinumerous with holey cyclically symmetric lozenge tilings of a hexagon. We establish a bounded version of a generalization of this identity. Further, we provide combinatorial interpretations of both sides of the identity. The ultimate goal would be to construct a combinatorial proof of this identity (possibly via an appropriate variant of the Robinson-Schensted-Knuth correspondence) and its unbounded version, as this would improve the understanding of the mysterious relation between alternating sign trapezoids and plane partition objects.

Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

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 solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$-groups with Kantor familes, by showing that finite groups K with a Kantor family $(\mathcal {F},\mathcal {F}^*)$ having distinct members $A, B \in \mathcal {F}$ such that $A^* \cap B^*$ is a central subgroup of K and the quotient $K/(A^* \cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members of $\mathcal {F}$ are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order $(t,t)$ (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order $(t,t)$ that we carry out in the present paper.

Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$.