Papers

For a partially ordered set ${\mathbb{P}} = (X,\leq)$ there exist hypergraphs where the vertices are the set of ordered tuples of either all incomparable elements of ${\mathbb{P}}$ or all the critical pairs of ${\mathbb{P}}$, and the edges are formed by the duals of either all the alternating cycles of ${\mathbb{P}}$ or all the strict alternating cycles of ${\mathbb{P}}$. The weak chromatic numbers of these hypergraphs are all equal to the order dimension of ${\mathbb{P}}$. Here are established upper bounds on the number of strict alternating cycles a poset ${\mathbb{P}}=(X,\leq)$ can have in terms of $n = |X|$, the cardinality of the groundset of ${\mathbb{P}}$, and the width $w$ of ${\mathbb{P}}$. These bounds also apply to the number of hyperedges of the associated hypergraph ${\mathcal{H}}^s(\mathbb{P})$, with incomparable pairs as vertices and strict alternating cycles dual to its hyperedges.

The Alon–Frankl–Lovász theorem determines the chromatic number of Kneser hypergraphs; equivalently, it gives the sharp minimum size of a monochromatic matching in every edge-colouring of a complete uniform hypergraph. Its known general proofs are topological. We introduce a topology-free structural framework. It reduces every colouring of a pseudorandom $t$-graph, with only $o(n)$ loss in the largest monochromatic matching, to a colouring of $K_n^{(t)}$ whose vertex set has at most $r$ parts and whose edge colours depend only on intersection profiles. Together with a stability analysis at the critical scale, we prove an exact robust form: there exists $c=c(r,t)>0$ such that, if a $t$-graph satisfies $\delta(\mathcal G)\ge(1-c)\binom{n-1}{t-1}$, then every $r$-colouring of $\mathcal G$ contains a monochromatic matching of the exact optimal size for all sufficiently large $n$. This gives a topology-free proof of the Alon–Frankl–Lovász theorem for large $n$, a sparse random transference theorem, and the exact value predicted by Meunier's stable Kneser conjecture throughout the range covered by our robust AFL theorem. We further develop the framework for Ramsey graph tilings. For a graph $H$, let $Rt_r(H;K_n)$ be the minimum, over all $r$-edge-colourings of $K_n$, of the largest monochromatic $H$-tiling. We prove $$ Rt_r(H;K_n)=(\beta_{r,H}+o(1))n, $$ where $\beta_{r,H}$ is effectively computable from finitely many rational linear programs depending only on $H$ and $r$. An additional multipartite Ramsey argument is needed to reconstruct a consistent coloured template. This gives an effective asymptotic solution to the multicolour Ramsey-tiling problem, extending the classical two-colour theorem of Burr, Erdős and Spencer. We also determine explicit constants for several natural families.

We decompose a symplectic column by detaching its maximal nonempty intervals that are fixed by the parity involution, this is the part of the symplectic column that is fixed by that involution, and the remaining part consisting of the subword whose image under the parity involution is disjoint of the symplectic column. The parity involution swaps an even number with the previous odd number and an odd number with the next even number. That decomposition is uniquely determined by the symplectic column and the symplectic conditions satisfied by those pieces pack the needed information to explicitly write the inverse of the reduction map in the quantum Littlewood-Richardson bijection. The tools provided here can be combined with the composition of the inverses of the several maps in which the reduction map decomposes, given by Watanabe, namely, among them, combinatorial $R$-matrices and reduction maps on shorter symplectic columns.

We determine the full graded Betti tables of graph families that subsume several classes studied recently in the literature, namely the generalized multiple complete split-like graphs and the generalized clique-star graphs with arbitrary clique block sizes. The method combines Hochster's formula with a precise decomposition of the associated independence complexes into disjoint unions of simplices and iterated joins of discrete complexes. This reduces every graded Betti number to an explicit coefficient extraction formula and yields closed expressions for the linear strand, higher strands, Hilbert series, regularity, projective dimension, and extremal Betti numbers. In particular, we prove a sharp criterion for $2$-linear resolution and identify the regularity corner in terms of the number of nontrivial clique blocks. As applications, we recover and extend earlier results on equal-block split-like graphs, obtain complete formulas for pineapple graphs, and derive consequences for power graphs of cyclic groups, elementary abelian groups, and prime-power dihedral groups.

The study of $(d+1)$-uniform set systems with VC-dimension at most $d$ links the Erdős–Ko–Rado theorem with VC-dimension. But already in 1997, Ahlswede and Khachatrian showed that this is not the right extension of the Erdős–Ko–Rado theorem. In 2025, Chao, Xu, Yip and Zhang proposed the uniform witness conjecture as a possible right extension: for $0\le s\le d$, if every set of a $(d+1)$-uniform family has a missing trace of the same fixed size $s$, then the family should have size at most $\binom{n-1}{d}$. They proved the conjecture when $s=d$, and when $s=1$ and $n$ is large. Very recently, Chao, Xu and Zakharov proved the conjecture when $s\le \frac{d}{2}$ and $n$ is large. We fill in the missing half of the picture, although the picture is not the one suggested by the conjecture. More precisely, for $d\ge 4$ and $\left\lceil \frac{d+2}{2}\right\rceil\le s\le d-1$, we construct such a family $\mathcal{F}\subseteq\binom{[n]}{d+1}$ with $|\mathcal{F}|=\binom{n-1}{d}+\binom{n-2(d+1-s)-2}{2s-d-2}$ for every $n\ge2(d+1)$, thereby disproving the uniform witness conjecture.

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs – where both adding and deleting any edge creates an induced copy of $H$ – appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.

Let $D$ be an oriented graph, and let $f_T(D)$ denote the number of copies of $D$ in a tournament $T$. We say that $D$ is converse invariant if $f_T(D)=f_T(\overline D)$ for every tournament $T$, where $\overline D$ is obtained from $D$ by reversing all arcs. Ai, Gutin, Lei, Yeo, and Zhou introduced a digraph polynomial for studying this property and conjectured that an orientation of a tree of maximum degree at least $3$ is converse invariant if and only if it is self-converse or can be obtained recursively by bridge-mirroring from an orientation of a path. We disprove this conjecture. More precisely, we characterize converse-invariant orientations of trees of diameter four and exhibit non-self-converse examples that do not arise from the recursive bridge-mirroring construction. To prove the classification, we introduce a multilinear polynomial $P_D$ encoding the difference $f_T(D)-f_T(\overline D)$ over all tournaments $T$, and we give a coefficient formula for $P_D$ as a signed sum over copies of subgraphs of the underlying graph of $D$. This polynomial method yields parity obstructions, gives new proofs that oriented paths and cycles are converse invariant, and provides the main tool for the diameter-four classification.

A convexity space is an ordered pair $(V,\mathcal{C})$, where $V$ is an arbitrary set and $\mathcal{C}$ is a family of subsets of $V$, called convex, which contains $\{\emptyset,V\}$ and is closed under intersections and nested unions of its elements. For any $S\subseteq V$, the convex hull of $S$ is the inclusion-wise minimum convex set $C\in \mathcal{C}$ such that $S\subseteq C$. For a convex set $C\in \mathcal{C}$, an element $p\in C$ is an extreme of $C$ if $p$ does not belong to the convex hull of $C\setminus\{p\}$. A convexity $\mathcal{C}$ defined over $V$ is a convex geometry if any convex set $C\in \mathcal{C}$ is the convex hull of its extreme elements. Given an oriented graph $D = (V,A)$, the family $\mathcal{C}$ of subsets of $V$ is the $\overrightarrow{P_3}$-convexity defined over $D$ if $\mathcal{C}$ is formed by all (convex) sets $C\subseteq V$ such that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ with $\{u,w\} \subseteq C$, while in the $\overrightarrow{P_3^*}$-convexity defined over $D$, we have that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ such that $\{u,w\} \subseteq C$ and $(u,w)\notin A$. In this work, we present necessary and sufficient conditions over an oriented graph $D$ so that the $\overrightarrow{P_3}$-convexity over $D$ is geometric, or the $\overrightarrow{P_3^*}$-convexity over $D$ is geometric. While the first case implies a polynomial-time algorithm to decide whether the $\overrightarrow{P_3}$-convexity over $D$ is a geometric, we show that it is coNP-complete to decide whether the $\overrightarrow{P_3^*}$-convexity over $D$ is a convex geometry. We also present a family termed acyclic indifference oriented graphs and demonstrate that deciding whether the $\overrightarrow{P_3^*}$-convexity in this class is geometric can be solved in polynomial-time.

We prove that for all $c\in\mathbb N$ and nonnull graphs $H_1,\ldots,H_t$, there exists $n\in\mathbb N$ such that if $G$ is a $c$-edge-colored complete graph with no monochromatic induced copy of the complete join of $H_1,\ldots,H_t$, then $V(G)$ is the union of $n$ sets $V_1,\ldots,V_n$ such that within each set $V_j$ with $|V_j|\neq 1$, the edges of some color form a graph that excludes at least one of $H_1,\ldots,H_t$ as an induced subgraph. In fact, the same holds even if the colors overlap, and with a different list of graphs $H_1,\ldots,H_t$ assigned to each color. When $H_1,\ldots,H_t$ each have a single vertex, this is Ramsey's theorem, and when $c=2$, this is the "excluding pairs of graphs" theorem of Chudnovsky, Scott, and Seymour.

We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles. Conversely, for every pair $(\alpha,\beta)\in[0,1]^2$ of irrational numbers in the same quadratic number field, we construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling whose density of vertical stripes is $\alpha$ and density of horizontal stripes is $\beta$.

Fix an abelian group $A$, a graph $G$, and nowhere-zero $A$-flows $f'$ and $f''$ on $G$. Now $f'$ and $f''$ are \emph{$A$-flow-adjacent} if there exists a cycle $C$ in $G$ such that $f'(e)-f''(e)=0$ for all edges $e\notin E(C)$. And $f'$ and $f''$ are \emph{$A$-flow-equivalent} if there exists a sequence $f_0,\ldots,f_s$ of $A$-flows such that $f_0=f'$, $f_s=f''$, and $f_i$ and $f_{i-1}$ are $A$-flow-adjacent for all $i\in[s]$. Given a group $A$, we seek conditions on a graph $G$ such that all $A$-flows on $G$ are pairwise $A$-flow-equivalent; in this case, we say that $G$ is \emph{$A$-flow-connected}. Analogously, we define $k$-flow-connectedness for nowhere-zero (integer) $k$-flows. The notions of $A$-flow-connectedness and $k$-flow-connectedness were first investigated by Esperet et al., who showed, among other results, that every $2$-edge-connected graph is $A$-flow-connected whenever $A=\mathbb{Z}_2^8$ or $|A| \ge 1.15\times 10^{694}$. In this paper, we first characterize the graphs that are $\mathbb{Z}_3$-flow-connected and that are $3$-flow-connected. We show that every 2-edge-connected graph is $A$-flow-connected if and only if this is true for every 2-edge-connected cubic graphs. We show that all cubic bipartite graphs are $\mathbb{Z}_4$-flow-connected, and construct other cubic graphs that are and are not $\mathbb{Z}_4$-flow-connected. We conjecture that every Eulerian graph is $k$-flow-connected and $A$-flow-connected whenever $k$ or $|A|$ is even; and provide evidence for this conjecture. Finally, we consider $4$-edge-connected graphs $G$. Here, we show that $G$ is $A$-flow-connected whenever $|A|\ge 5.3\times 10^6$.

The highest rank of a string C-group representation of the alternating group $A_n$ is known for each $n$, but no self-dual representations attaining this highest rank are known when $n > 12$. Motivated by computational results for alternating groups of small degree, we examine a vertex-gluing construction for permutation representation graphs. We establish conditions under which gluing two string C-groups produces another string C-group, and use this construction to obtain infinite families of self-dual representations of alternating groups. In particular, for every $n = 4m+3 \geq 15$, we construct $\left \lfloor \frac{n+9}{8} \right \rfloor$ distinct self-dual string C-groups of rank $2m$ isomorphic to $A_{n}$. These representations have rank one below the maximum possible rank of string C-group representations for $A_n$, and to the authors' knowledge are the highest-rank self-dual representations currently known for alternating groups.

We give a short, self-contained linear-algebra proof of a bound of Hegedüs [Australasian Journal of Combinatorics, 2026; arXiv:2409.07877]: if all pairwise Hamming distances in a family of subsets of $\{1,\ldots,n\}$ equal a fixed value $\lambda\ne(n+1)/2$, then the family has at most $n$ members. Our proof uses the same Gram matrix as in Hegedüs's argument, but reads its eigenvalues in place of its determinant, and keys off of a single fact about vectors of equal norm and equal pairwise inner product. That fact applies verbatim over an alphabet of size $q$, where it yields the bound $n(q-1)$ for $\lambda\ne\bigl((q-1)n+1\bigr)/q$ – the corrected form of a conjecture of Hegedüs, recently established by Hu, Huang, and Yu [arXiv:2504.07036].

Adapting the construction disproving the sum-product conjecture over $\mathbb R$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A \subseteq \mathbb R$ such that $$|AA+A+A| \ll |A|^{2-c}.$$ As a corollary, all of the sets $A+A$, $AA$, $(A+1)(A+1)$, $A(A+1)$ and $AA+A$ are of size $O(|A|^{2-c})$ for this construction.

The binary Type~I self-dual [26,13,6] code is classical. We give what appears to be the first direct non-computational proof of its uniqueness. We first determine the weight enumerators of the code and its shadow. Degree-one harmonic MacWilliams identities supply the required 1-designs. Elementary intersection counts then show that the two minimal half-shadows each contain 13 words; these words label the 26 coordinates as 13 points and 13 lines, and the two shadow classes become the point-stars and line-stars. From this structure we give two uniqueness proofs: one reconstructs the projective plane of order 3 and the plane code, including the full automorphism group PGL(3,3):2; the other deletes an intrinsic flag, obtains the odd Golay code together with a deep hole coset, and reconstructs the length 26 code and the size of the automorphism group from this coset datum. Thus the natural length-24 object behind the code is the odd Golay code together with its unique orbit of deep hole cosets.

Nilpotent Lie groups with left-invariant metrics provide nontrivial examples of Ricci solitons. Some typical examples are given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs and the class of nilpotent Lie algebras obtained from finite acyclic quivers. In this paper, we generalize the construction of nilpotent Lie algebras that are algebraic Ricci solitons obtained from finite acyclic quivers. We use some special ordered sets to construct nilpotent Lie algebras, which can also be obtained from some special quivers with relations. A transitively and antisymmetrically ordered set (or TAOS, for short) is a set together with a binary relation that is transitive and antisymmetric. Utilizing the concept of incidence algebras of TAOSs, we construct nilpotent Lie algebras. We modify the method introduced by Mizoguchi and Tamaru \cite{MR4941781} and use it to show that the nilpotent Lie algebras with arbitrarily high degrees of nilpotency obtained from some special finite transitively and antisymmetrically ordered sets, called array TAOSs, are algebraic Ricci solitons. We also give some generalizations of this result, which yield more nilpotent Lie algebras that are algebraic Ricci solitons. Moreover, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons.

We introduce a new family of balance indices for phylogenetic networks: the $H_\alpha$ indices, where $\alpha$ is a positive real number. This family includes the $B_2$ index as a special case ($\alpha = 1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_\alpha$ indices share many structural properties with the $B_2$ index, most notably a "grafting property" that makes it possible to express the $H_\alpha$ index of a network in terms of the $H_\alpha$ indices of its biconnected components. These properties allow us to identify networks that minimize / maximize $H_\alpha$ for various classes of phylogenetic networks, and to study its distribution for several models of random trees and networks (in particular, Galton-Watson trees and binary Markov branching trees, with a focus on the Yule and PDA models). Finally, we show how local limits can be used to analyze the asymptotic behavior of $H_\alpha$ for large trees and networks, and we obtain general results for the moments of $H_\alpha$ for a broad class of random phylogenetic networks known as blowups of Galton-Watson trees.

We study a vector-space analogue of the Erdős Matching Conjecture. Let $m_q(n,k,s)$ denote the maximum cardinality of a family of $k$-dimensional subspaces of an $n$-dimensional vector space over $\mathbb F_q$ with no $s+1$ members whose sum is direct. Two natural constructions provide lower bounds. The first consists of all $k$-subspaces contained in a fixed $((s+1)k-1)$-dimensional subspace; the second consists of all $k$-subspaces that intersect a fixed $s$-dimensional subspace nontrivially. These constructions motivate the following vector-space analogue of the Erdős Matching Conjecture: for all $n\ge (s+1)k$, $$m_q(n,k,s)=\max\left\{\genfrac{[}{]}{0pt}{}{(s+1)k-1}{k}_q,~\genfrac{[}{]}{0pt}{}{n}{k}_q-q^{ks}\genfrac{[}{]}{0pt}{}{n-s}{k}_q\right\}.$$ We prove this conjecture when $k=2$, when $n=(s+1)k$, and when $n$ is sufficiently large. In particular, the case $k=2$ may be viewed as a vector-space analogue of the Erdős–Gallai theorem. In the large-$n$ range, we also prove a Hilton–Milner-type stability theorem, determining the largest nontrivial families with this property. Finally, we connect this problem with $t$-cover-free families in vector spaces and determine their extremal number up to a lower-order term, extending a recent result of Shan and Zhou for the special case $t=2$. The proofs combine Lovász's minimax theorem for matroid matchings, a high-dimensional Hoffman bound for uniform hypergraphs, and packing-design arguments in vector spaces.

We show that random linear codes (RLCs) possess nearly optimal discrepancy-type properties in a broad range of settings. Our main results are two general discrepancy theorems: one controls all translates of a fixed test, and the other controls large families of Fourier-pseudorandom tests. Two motivating examples follow: First, RLCs behave essentially like unstructured random codes for list-decoding from errors above capacity. More precisely, an RLC $C\subseteq \mathbb{F}_q^n$ of rate $1 - \frac{1}{n}\log_q|B_\rho| + \varepsilon$, where $|B_\rho|$ is the volume of a radius-$\rho$ Hamming ball in $\mathbb{F}_q^n$, satisfies $|C \cap B| = (1\pm o(1)) \frac{|C|\cdot |B|}{q^n}$ simultaneously for all radius-$\rho$ Hamming balls $B$ with high probability. This vastly generalizes the previously best known fact that RLCs of this rate have covering radius at most $\rho n$ with high probability (Blinovsky, 1987). Second, over prime fields, RLCs behave essentially like unstructured random codes for zero-error list-recovery, and list-recovery from erasures, above capacity. More precisely, for a prime $q>2$ and input list size $2\leq \ell\leq q-1$, an RLC $C\subseteq \mathbb{F}_q^n$ of rate $1-\log_q \ell+\varepsilon$ will satisfy $|C \cap S| = (1\pm o(1)) \frac{|C|\cdot \ell^n}{q^n}$ simultaneously for all combinatorial rectangles $S=S_1\times S_2\times\cdots\times S_n$, where $|S_i|=\ell$ for all $i$, with high probability. An analogous result also holds when we can bound $|S_i|$ only for some of the $i$'s. We use this to show the abundance of locally leakage-resilient $n$-party linear ramp secret sharing schemes with any linear reconstruction threshold and sublinear threshold gap $O(n/\log n)$ over fields of polynomial size $q=\Theta(n^\gamma)$ for a constant $\gamma\in(0,1/5)$. Prior work was stuck at reconstruction thresholds above $n/2$ for both threshold and ramp schemes.

This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph $G$ and its associated quotient graph $G/\pi$, induced by an equitable partition $\pi$. Through the shunt decomposition of $G$, we derive an explicit relation between the shift operator of the parent graph $G$ and that of its quotient graph $G/\pi$. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on $G$ if and only if it occurs on $G/\pi$. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph $C_{n}$ at time $k = n/2$, and lift the result to the parent graph $C_{2n}$ via the equitable partition $\pi$. We further show that if an equitable partition $\pi$ of $G$ induces a quotient isomorphic to $K_n^{\circlearrowleft}$, the complete digraph on $n$ vertices with a loop at every vertex, then PST occurs at step $k = n$, and the walk is periodic at $k = 2n$. This framework is applied to two families of graphs, which are the complete bipartite digraph $K_{n,n}^{\rightleftharpoons}$ and the circulant graph $\operatorname{Circ}(2n, S)$, where $S$ consists of all odd residues modulo $2n$ and $n = 2^s$ for some $s \geq 1$, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.

Let $p$ be an odd prime and let $S\subseteq \Z_p$ be symmetric with $0\notin S$. Let $\Cay(\Z_p,S)$ be the undirected Cayley graph on $\Z_p$ in which $x$ and $y$ are adjacent if and only if $x-y\in S$. For $1\le \ell\le (p-1)/2$, define \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=\max\{|S|: S=-S,\ 0\notin S,\ \Cay(\Z_p,S)\text{ contains no }C_{2\ell+1}\}. \] Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\ell+1$, then $\ex_{\Cay}(C_{2\ell+1},\Z_p)=0$, while if $p>2\ell+1$, then \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=2\floor{\frac{p+2\ell+1}{2(2\ell+1)}}. \] The proof combines a sharp additive zero-sum odd-girth argument with weak odd pancyclicity to transfer the result from odd-girth exclusion to fixed odd-cycle exclusion. We also give a canonical extremal family, an exact extremality criterion in terms of odd zero-sum avoidance, and an example showing that extremizers need not be dilates of the canonical construction.

The $n$-queens completion threshold $qc(n)$ is the largest integer $k < n$ such that any placement of $k$ mutually non-attacking queens on an $n \times n$ chessboard can be completed to an $n$-queens configuration by adding $n - k$ queens. For all sufficiently large $n$, we improve the previously best-known upper bound on $qc(n)$ from $qc(n) \leq 0.241n$ to $qc(n) \leq 0.216n$, by constructing a non-completable partial configuration of fewer than $0.216n$ queens.

We study a general class of self-repelling polymers on $\mathbb Z^2$, including the simple random walk, the self-avoiding walk and the repulsive Domb-Joyce model, in the presence of a constant force field acting on each monomer. Conditioning the polymer to have fixed length and fixed endpoints, we identify the limiting free energy and prove that typical trajectories concentrate exponentially near a deterministic macroscopic shape. This shape is characterized as the unique minimizer of a variational problem and can be interpreted as a geodesic of a height-dependent Finsler metric. We also analyze two limiting regimes with universal features: for small field strength, in the symmetric case, the geodesic is close to a classical catenary, while for large field strength it converges to a universal polygonal shape governed by the nearest-neighbor lattice constraint.

Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \[ \dim L\leq \frac{t+m-2}{m}. \] Here $\ell_1,\ldots,\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\ell_1^m,\ldots,\ell_t^m$ form a circuit, and $L=\Span\{\ell_1,\ldots,\ell_t\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\dim L)$; in particular, the affine bound itself is sharp and the optimal leading constant in Bukh's question is $1/m$. The proof uses a coding-theoretic translation: the coefficient row space of the powers is the $m$-th Schur power of the coefficient code, and the minimality hypothesis makes this Schur power a full-support hyperplane to which the Schur-product Kneser theorem of Mirandola and Zémor applies. The same method yields flat-concentration and interpolation criteria, a Cayley–Bacharach lower bound, Segre–Veronese and positive-characteristic variants, and Hilbert-function constraints for equality and near-equality in Veronese circuits.

By bridging two classification results – the Esterov–Gusev classification of tuples of lattice polytopes of mixed volume one, and Fink's characterization of Bergman fans – we formulate a classification conjecture describing when the stable intersection of a tropical fan F with the tropicalization Trop(X) of a subvariety of T^n is a reduced point. Our main results establish this conjecture in three fundamental cases – unmixed sequences, hypersurface complete-intersection cycles, and tropical 2-cycles – and develop several tools intended for the general case.

Given a graph class $\mathcal{G}$, the limiting density of $\mathcal{G}$ is defined as $\delta(\mathcal{G})=\lim_{n\to\infty} \mathsf{ex}(\mathcal{G},n)/n$ where $\mathsf{ex}(\mathcal{G},n)$ is the maximum number of edges of a graph in $\mathcal{G}$ on $n$ vertices. The limiting density $\delta(\mathcal{G})$ is known to be a rational number when $\mathcal{G}$ is a minor-closed graph class. For every $\delta\in[0,\frac{3}{2})$, we prove that the set of $\subseteq$-minimal minor-closed graph classes with densities $>\delta$ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs $\mathcal{Z}$, of total size $n$, either outputs the value of $\delta(\mathsf{excl}(\mathcal{Z}))$ or reports that $\delta(\mathsf{excl}(\mathcal{Z}))\geq \frac{3}{2}$, where $\mathsf{excl}(\mathcal{Z})$ is the class of graphs excluding the graphs in $\mathcal{Z}$ as minors. The algorithm runs in $2^{\mathsf{poly}(n)}$ time.

We develop a unified framework for the exact enumeration and asymptotic analysis of the three most studied classes of phylogenetic networks: tree-child (TC), reticulation-visible (RV) and orchard networks, whose cardinalities satisfy the strict ordering $|\mathrm{TC}_{\ell,k}|<|\mathrm{RV}_{\ell,k}|<|\mathrm{Orch}_{\ell,k}|$ for reticulation number $k\geq2$ (with $\mathrm{TC}\subsetneq\mathrm{RV}$ and $\mathrm{TC}\subsetneq\mathrm{Orch}$, while $\mathrm{RV}$ and $\mathrm{Orch}$ are incomparable as sets). Using the Chang–Fuchs structural theorem, we derive a two-level master functional equation for the RV bivariate generating function and obtain exact closed-form identities for the differences $\Delta_k(\ell):=|RV_{\ell,k}|-|TC_{\ell,k}|$ for $k=2,3$, with the asymptotic universality $\Delta_k(\ell)/|TC_{\ell,k}|\sim k!/\ell$. For orchard networks, we prove a universal hypergeometric law that resolves the exact enumeration problem for all $\ell$: the column generating function $F_\ell(v)$ is rational with denominator $D_\ell(v)=\prod_{j=2}^\ell X_j(v)$, where \[ X_\ell(v) = \sum_{k=0}^{\lfloor\ell/2\rfloor}(-1)^k\, \frac{\ell!}{(\ell-2k)!\,k!}\,v^k \] is the matching polynomial of the complete graph $K_\ell$ and a rescaled Jacobi polynomial. This immediately resolves the intractable $\ell=9$ case: $D_9$ has degree 20, dominant growth rate $\approx40.73$, and all spectral roots are positive real. A complete enumeration table is provided extending the published data of Cardona, Ribas and Pons.

Toric arrangements of maximal rank have been studied by the author in a paper that shows how the complement manifold of these arrangements is diffeomorphic to that of centered ones. In this work we turn our attention to toric arrangements of rank one, namely parallel toric arrangements. Our aim is to prove, by means of basic arguments of cohomology theory, that the Euler-Poincaré characteristic of the complement manifold of parallel toric arrangements can be computed in terms of those of the complement manifolds of the singular subtori that compose the arrangement.

Additive codes over $\mathbb{F}_{q^h}$ generalize linear codes by relaxing linearity over the alphabet while retaining linearity over the subfield $\mathbb{F}_q$. In this paper, we introduce minimal additive codes and we initiate their study from a geometric perspective. We define the concept of additive strong blocking sets, a class of $h$-projective systems whose union forms a strong blocking set. We establish a one-to-one correspondence between equivalence classes of nondegenerate minimal additive codes and equivalence classes of additive strong blocking sets. We also compare this framework with the theory of outer strong blocking sets, showing that the latter arises as a special case. Finally, we provide constructions and existence results for minimal additive codes, and derive upper, lower, and asymptotic bounds on their minimum length.

In this paper, we study the $\alpha$-Ricci curvature and the Lin-Lu-Yau Ricci curvature on simple, connected, and locally finite graphs. For regular graphs, we introduce a combinatorial construction of optimal transport plans realizing the 1-Wasserstein distance and use it to derive exact formulas for the $\alpha$-Ricci curvature and the Lin-Lu-Yau Ricci curvature. This yields a combinatorial proof of the known curvature formulas. Furthermore, for non-regular graphs, we characterize conditions on the size of common neighborhoods that guarantee either non-negative or vanishing Lin-Lu-Yau Ricci curvature.