Papers

We recall the concept of the dimension of a finite poset $P$, and the longstanding conjecture that for all finite nonempty posets $P$ and $Q$, $\dim(P\times Q)\geq\dim(P)+\dim(Q)-2.$ We then note two other plausible inequalities, either of which would imply that one. In the final section, writing $P\preccurlyeq P'$ if, for all $Q,$ $\dim(P\times Q)\leq\dim(P'\times Q),$ and writing $P\approx P'$ if $P\preccurlyeq P'$ and $P'\preccurlyeq P,$ we note some results and questions concerning these relations.

We prove that there is a density-one set of positive integers for which the products of all distinct consecutive blocks are distinct, answering a question of Erdős and Graham. The construction retains or rejects entire interiors of consecutive-prime gaps and is therefore stable under passage to longer prefixes. Uniform affine-curve estimates control the witnesses with no previous rejected gap, and a scale-contracting forest controls every remaining witness unconditionally.

We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $\sigma(n) \ge \lceil n(n-2)/6 \rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\lfloor (n-1)^2/4 \rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.

Let $F(n)$ be the largest size of a set $A\subseteq[1,n]$ such that $a\nmid bc$ whenever $a,b,c\in A$ and $a\notin\{b,c\}$, with $b$ and $c$ allowed to coincide. We prove \[ F(n)=\pi(n)+\left(\frac{27}{2}+o(1)\right)\frac{n^{2/3}}{(\log n)^2}. \] This determines the second-order constant conjectured by Erdős; the upper bound keeps the leading constants in his multiplicative basis, while the lower bound packs scale-separated prime triples by proper edge-colourings.

The existence, several properties, and constructions of Generalized Weighing-Hadamard (GWH) matrices over finite fields are addressed in this work. We study the subset of invertible GWH matrices and show that it forms a group under matrix multiplication. Besides that, we introduce a strong notion of equivalence between such matrices, defined via orthogonal transformations, and further prove that the corresponding quotient group by the subgroup of orthogonal matrices is abelian. Finally, we discuss some applications of these matrices in coding theory

We develop a geometric framework in which the token dynamics of a transformer are modeled by a system of interacting particles on a Riemannian manifold $\mathcal M$, the attention mechanism being encoded by a time-independent two-body interaction law, that is, a section of the pullback bundle $\pi_2^{*}(T\mathcal M)$ over $\mathcal M\times\mathcal M$. Within this framework we isolate two features that a family of interaction laws must possess in order to model language: it must realize generic nonlocal and nonreciprocal forces, and it must parametrize vector fields on a high-dimensional manifold efficiently. We show that both features are achieved simultaneously in a transformer model. Our main theorem produces a finitely parametrized family of interaction laws that is universal: it realizes an arbitrary prescribed attention digraph. Moreover, we show that the cost of realizing a given attention digraph is governed by two combinatorial invariants of the digraph, namely its biclique cover number, which we identify with the least number of hubs in a hub extension, and its hub-chromatic index.

Cops and Robbers is a well-studied pursuit-evasion game that provides insights into graph theory and theoretical computing. A central question is determining the minimum number of cops required to capture the robber, known as the cop number. We focus on reproducing an algorithm proposed by Petr, Portier, and Versteegen in 2022, which efficiently determines whether a graph is $k$-copwin. This paper presents a Python implementation of the $k$-copwin algorithm. In this work, we present our implementation in detail, clarify key aspects of the algorithm, and discuss its implications for future practical deployments.

The commutator of a pair of vector fields on a graph is not a vector field in general, but rather a second order differential operator. We investigate this departure from the classical case of vectors fields on a manifold by examining the geometry of balls of radius two, concentrating on the set of paths of length two connecting a given vertex with the center of the ball. There is a natural surjection from the space of sections of the second tangent bundle to the space of second order differential operators whose kernel reflects the geometry of these balls. Using this map we draw several conclusions about second order differential operators including canonical forms, formulas for their adjoints, and a necessary and sufficient condition for a commutator to be a vector field.

We prove Sabidussi's compatibility conjecture. Let $G$ be a finite connected multigraph in which every vertex has even degree and the minimum degree is at least four, and let $T$ be an Euler tour of $G$. The edges of $G$ can be partitioned into circuits (connected $2$-regular subgraphs) so that no circuit contains two edges used consecutively anywhere in $T$. In fact, the edges can be four-coloured so that every such pair receives different colours and every colour class has even degree at every vertex. We use a counting argument based on the Chevalley-Warning theorem to show that a four-colouring with the required properties exists. Splitting each colour class into circuits then gives the desired compatible decomposition. Formalization in Lean 4 is also available in the author's github.

We consider the space of $(n-1)$-colored graphs on a fixed set of $N$ vertices. Each edge position of the complete graph $K_N$ has $n$ possible states: the absence of an edge and $n-1$ colors. This gives a natural identification of the space of such graphs with the tensor power $(\mathbb C^n)^{\otimes m}$, where $m=\binom N2$, and defines on it the diagonal action of the Lie algebra $\mathfrak{gl}_n$, and, after restriction, the action of $\mathfrak{sl}_n$. For a fixed profile $\alpha=(\alpha_0,\dots,\alpha_{n-1})$, we consider the graph $J(m;\alpha)$ whose vertices are colored graphs of this profile and whose adjacency is defined by a single exchange of states in two edge positions. This graph is the transposition graph on the set of words with fixed profile, also known as the multislice. The main result is an expression of the adjacency operator in terms of the root operators of $\mathfrak{sl}_n$ and a derivation of its spectrum by means of the quadratic Casimir operator of $\mathfrak{gl}_n$ and the Schur–Weyl decomposition. It is proved that the adjacency operator belongs to the center of the algebra $\End_{S_m}(\mathcal C_\alpha)$. The contribution of each spectral block to the multiplicity of the corresponding eigenvalue is described in terms of a Kostka number and the dimension of a Specht module. For $n=2$, one obtains the classical Johnson graph and its known spectrum. As applications, a formula for the valency is established, connectivity is proved, the Hoffman bound for independent sets is obtained, and the three-state case is considered in detail; in this case the natural symmetrized subspace realizes the module $\Sym^m(\mathbb C^3)$.

We give an algebraic branching program of polynomial size which computes Cayley determinant of right quantum matrices. This is a rare example of an efficient computation of a noncommutative determinant, and the first such example for quantum groups. We extend the results to the $q$-Cayley determinant of $q$-right quantum matrices, as well as to their multiparameter generalization. The proofs are entirely combinatorial, as we relate Cayley, Moore and Valiant determinants using bijections/involutions on words. We then employ the celebrated determinant construction of Mahajan and Vinay (SODA'97), to obtain the results.

A graph is called resistance nonnegative (RN), respectively resistance positive (RP), if it admits positive edge weights such that all vertex resistance curvatures are nonnegative, respectively positive. In this paper, we study the structure of RN and RP graphs in relation to toughness, traceability, and Cartesian products. First, we disprove a conjecture of Fiedler and answer a question of Devriendt in the negative by constructing, for every $n\ge 11$, an $n$-vertex $1$-tough graph that is not RN. Second, we show that RP graphs need not be traceable by proving that the Thomassen $34$-graph is RP but not traceable. Finally, we resolve a conjecture of Devriendt on grid graphs by proving that all Cartesian products of paths are RN.

For integer vectors R,S let A(R,S) denote the class of (0,1)-matrices with row sum vector R and column sum vector S. Its interchange graph G(R,S) has A(R,S) as its vertex set, two matrices being adjacent when they differ by a single 2 x 2 interchange. Brualdi asked whether G(R,S) is Hamiltonian for every R,S. We prove the stronger statement that G(R,S) is maximally Hamiltonian: Hamilton-laceable when bipartite, and Hamilton-connected when not. The proof is a structural induction on the number of matrices in the class, organized by the structure theory of interchange graphs. Deleting inactive lines and splitting invariant positions expresses any class as a Cartesian product, reducing the argument to the prime factors. The bipartite classes are products of complete transposition graphs; we settle them together, without induction, by proving they are paired 2-disjoint-path-coverable and hence Hamilton-laceable, using a recent theorem of Coleman, Fischberg, Gong, Harrington and Wong on paired disjoint path covers. The non-bipartite classes divide into three cases: products assembled from smaller factors, a base of Johnson graphs and small classes, and the large prime classes, treated by a pivot-and-fiber construction whose line quotients are matroid base-exchange graphs. The complete argument has been machine-checked in the Lean 4 proof assistant from first principles together with seven cited results of the literature; the disjoint-path-cover results it imports are themselves proved within the formalization.

When $m = 1$, the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonholomorphic Jacobi forms. The rank is a statistic on partitions, and the higher Dyson systems, for $m \geq 2$, are a natural multivariable refinement of it, combining $m$ graded rank contributions. Unlike the classical case, these higher systems are not expected to fit the mock-modular framework, which raises the question of what analytic structure governs them. We show that their root-of-unity specializations carry a hidden elliptic structure. A finite $q$-difference recurrence produces an explicit polynomial obstruction to the expected index $m$ elliptic transformation law, and because the obstruction is finite, its partial fractions canonically determine finitely many Appell–Lerch correction terms that remove it. The corrected functions satisfy a twisted index $m$ elliptic law; a natural translation removes the twist, and their holomorphic finite parts admit finite theta decompositions. Thus, the natural analogue of Dyson's mock-modular phenomenon at higher $m$ is not mock modularity but a finite theta decomposition governed by an index $m$ elliptic transformation law. These results grew out of a human–AI collaboration, and the key new formulas were formalized and machine-verified in Lean/Mathlib by AxiomProver.

We show that every planar graph with $n \geq 3$ vertices admits a 4-coloring in which each color is used on fewer than $n/2$ vertices. This bound is the best possible. Moreover, such a coloring can be found in $O(n \log n)$ time. We also extend these results to five or more colors and to graphs on general surfaces.

We show that for every $\eta>0$ and sufficiently large $n$, there exists a poset of size $2^{(1+\eta)n/2}$ containing all the $n$-element posets as induced subposets. This improves a recent result of Bastide, Groenland and Nenadov. Our proof provides a labeling scheme preserving transitivity, inspired by the Boolean lattice. Among other tools, we use the Szemerédi Regularity Lemma.

How large can a set of mutually orthogonal quantum Latin squares (MOQLS) get? We show that a set of n - 2 MOQLS of order n is necessarily classical and construct large non-classical sets of MOQLS of orders that are prime powers, improving both the previously known lower and upper bounds.

For an integer $s \ge 2$, a subset $S \subseteq [n]$ is {\em $s$-stable} if $\min \{j - i, n + i - j\}\ge s$ for every $i,j \in S$ with $i<j$. Denote the set of all $s$-stable subsets of size $k$ of $[n]$ by $\binom{[n]}{k}_{s\text{-stable}}$. Schrijver proved in 1978 that whenever $n\ge 2k$, the chromatic number of the Kneser graph $\mathrm{KG}\big( \binom{[n]}{k}_{2\text{-stable}}\big)$ is $n - 2k +2$. Generalizing this result, Meunier conjectured in 2011 that $\chi\left( \mathrm{KG}\big( \binom{[n]}{k}_{s\text{-stable}} \big) \right)= n - sk +s$ for all $n\ge sk$. This conjecture was previously proven for all even $s$, for $s \ge 4$ and large enough $n$, and for $k=2$. We prove the conjecture in the cases $s=3$ and $n$ large enough, or $k=s=3$. To this end, we prove versions of the Hilton-Milner theorem for $s$-stable sets. We also present a topological approach towards Meunier's conjecture.

A claw-free cubic graph is a cubic graph with no induced subgraph isomorphic to $K_{1,3}$. The zero forcing process begins with an initial set $S$ of colored vertices. At each step, a colored vertex with exactly one uncolored neighbor forces that neighbor to become colored. If repeated applications of this rule color every vertex of $G$, then $S$ is called a zero forcing set. The minimum cardinality of a zero forcing set is the zero forcing number, denoted by $Z(G)$. In this paper, we answer three open questions posed by Davila and Henning concerning upper bounds on the zero forcing number of claw-free cubic graphs. We characterize the connected claw-free cubic graphs satisfying $Z(G)=\alpha(G)+1$, where $\alpha(G)$ is the independence number. In addition, we establish the improved upper bound $Z(G)\leq \frac{T}{2}+D+2$ for claw-free cubic graphs with Hamiltonian contraction multigraphs, where $D$ is the number of diamonds and $T$ is the number of triangles in $G$.

Amdeberhan, Shareshian, and Stanley recently proved that a function $\varphi$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\{1,\dots,2n\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\{1,\dots,2n\}$ with record composition $(\alpha_1,\dots,\alpha_\ell)$ is \[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2\alpha_j-1}E_{2\alpha_j-1}, \] where $s_j=\alpha_1+\dots+\alpha_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\dots a_{2n}$ (so $a_1>a_2<a_3>\dotsb$) lists the factor lengths obtained by cutting $a_1a_3\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\sec(\sqrt{t}\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.

Snir and Yuster [Discrete Appl. Math. 347 (2026) 160–171] asked for the least number $h(k)$ such that $k$ unrooted binary phylogenetic trees on the same $h(k)$ leaves always share a common quartet. We give a new upper bound for the $k$-tree version of the Maximum Agreement Subtree problem, namely an upper bound for the number of leaves, on which $k$ unrooted binary phylogenetic trees always share a common induced binary subtree on $n$ leaves, which is a four-times iterated exponential function. For $h(k)$, this implies a four-times iterated exponential upper bound. We also set an exponential lower bound for $h(k)$.

For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides Young diagrams at the intrinsic majority threshold. If the first row or first column contains more than half of the boxes, the Bernstein creation formula separates each rectangular-cycle character value into a hook constant and a remainder supported only on short cycles. The restrictions of the trivial and sign characters to the cyclic subgroup evaluate the constants exactly, whereas a uniform binomial contraction controls every distance from the boundary at once. If neither a row nor a column has a majority, Swanson's opposite-hook estimate gives a Catalan-scale lower bound for the dimension. The Fomin–Lulov rectangular-character estimate, together with uniform bounds for the truncated Möbius weights, then shows that the identity class dominates. Sundaram's plethystic identity consequently implies that $\prod_{r=1}^M(1-p_r)^{-1}$ is Schur-positive for every fixed $M$. We also record a degree-$12$ counterexample to a separate congruence-class product conjecture attributed to Richard Stanley.

We study a pinned form of graceful labeling. For full binary trees, we ask whether some deepest root-to-leaf path can carry the alternating extreme pattern $0,n-1,1,n-2,\dots$. Such a spine uses the extreme labels and largest differences, forcing all off-spine vertices and edges to use the middle labels and smaller differences, respectively. We prove this pinned-spine conjecture for comb full binary trees, verify it computationally for all rooted non-isomorphic full binary trees through order $23$, and give an example showing that a pinned-spine labeling cannot always be chosen as an $\alpha$-labeling. For spider trees, we prove a packing theorem for self-matched legs: pairwise disjoint legs based at hub label $1$, at least one of which contains label $0$, can be combined into a graceful spider, with unused labels attached as hub leaves. This yields graceful labelings for mixed-length spiders with sufficiently many leaves. We also report computations using a depth-first search ordered by largest unused differences and formulate the six-arm problem as an offset five-arm residual problem.

In this paper, we compute the Möbius function on the set of pseudo-Levi subgroups containing a fixed maximal torus of two families of finite groups, $GL_n(q)$ and $Sp_{2n}(q)$, for some natural number $n$ and a prime power $q$. The Möbius function for the set of pseudo-Levi subgroups of a finite group $G$ is important for an explicit evaluation of the formula expressing the decomposition of tensor products of characters, as proved in Theorem 2 of the paper ”Multiplicity of characters of finite reductive groups and Drinfeld doubles” [arXiv:2512.01432].

We introduce local flag algebras, a variant of Razborov's flag algebra framework in which densities are normalised by the maximum degree $\Delta(G)$ rather than the order $|G|$. The framework supports the same semidefinite-method machinery as the classical version, but is tailored to extremal problems that scale with the maximum degree. As an illustrative first application we bound the number of pentagons in a triangle-free graph $G$ as a function of $|G|$ and $\Delta(G)$.

Let $G$ be a digraph with vertex set $\{1,2,...,n\}$ and $H_{1},H_{2},...,H_{n}$ be $n$ digraphs. The generalized join digraph $\overrightarrow{G}=G[H_{1},H_{2},...,H_{n}]$ is a digraph obtained from $G$ by replacing each vertex $i$ with $H_{i}$ and for any $u\in V(H_{i})$ and $v\in V(H_{j})$, $(u,v)\in E(\overrightarrow{G})$ if and only if $(i,j)\in E(G)$. In this paper we express the number of oriented spanning trees in $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. Furthermore, we consider the number of oriented spanning trees with a fixed root in $\overrightarrow{G}$. First, we introduce the biclique-directed star transformation formula for counting oriented spanning trees with a fixed root in digraphs. Using it, we give the formula for the total number of oriented spanning trees with roots in a certain $H_{i}$ $(1\leq i \leq n)$ of $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. As applications, when each $H_{i}$ is a given digraph, the enumerative formulas for oriented spanning trees with a fixed root of $\overrightarrow{G}$ are derived from our work.

Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\ldots,s)\le cdr^{2}s^{r}\log r\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=\Theta(s^{2})$, and $f_{r}(d,s,\ldots,s)=\Theta_{d,r}(s^{r}\log{s})$ for every fixed $d,r$ with $d\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\ge4$. Beyond these sharp results, when $r\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\ldots,s)\le c_{d}rs^{r}\log(ers^{r})$ and $f_{r}(d,s,\ldots,s)\le c_{d}r^{d+2}s^{d+1}\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\ldots,s)>s^{\min\{r,d\}}$ for every $d\ge2$. Finally, we study two colored analogues. The direct Bárány–Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.

We give the conjectured solution for the generating function of $k$-convex polyominoes, enumerated by semi-perimeter. The solution was obtained from the analysis of enumeration data that we generated.

For a graph $G$, let $\Delta(G)$, $\omega(G)$, and $\chi(G)$ denote the maximum degree, clique number, and chromatic number of $G$, respectively. Let $P_n$ and $C_n$ denote the chordless path and chordless cycle on $n$ vertices, respectively. In this paper, we prove that every $(P_6,C_4)$-free graph $G$ with $\Delta(G)\ge 9$ and $\omega(G)<\Delta(G)$ is $(\Delta(G)-1)$-colorable.

The cover number of a graph by a graph class $\mathcal P$ is the least number of $\mathcal P$-graphs necessary to cover its edges. A classical theorem of Harary, Hsu and Miller gives an exact formula for the cover number by the class of graphs with chromatic number at most $k$. We investigate analogous questions for the case of the fractional chromatic number $\chi_f$ and the local chromatic number $\psi$. We prove that an analogous formula cannot hold in the case of the cover number by graphs of fractional chromatic number at most $\beta$, and find a lower and an upper bound, that gives rise to interesting asymptotic questions. We also investigate this cover number for small specific graphs. In the case of the cover number by graphs with local chromatic number at most $k$, we find an upper bound in terms of $\psi$, and a lower bound in terms of $\omega$.