Papers

For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai–Ramsey number $GR(G_1,\ldots,G_k)$ is defined analogously, with the colorings restricted to Gallai colorings (i.e., edge-colorings with no rainbow triangle). A copy of $P_3$ is called a cherry. Let $n_iP_3$ denote the disjoint union of $n_i$ cherries. Wu, Magnant, Nowbandegani, and Xia (Discrete Appl. Math., 2019) proposed two conjectures: \[ R(n_1P_3,\ldots,n_kP_3)=N\ \text{and}\ GR(n_1P_3,\ldots,n_kP_3)=N\,, \] where $N=2\max\{n_1,\ldots,n_k\}+\sum_{i=1}^kn_i-k+1$. We disprove the Ramsey conjecture and provide some sufficient conditions for determining the exact value of $R(n_1P_3,\ldots,n_kP_3)$. In contrast, we confirm the Gallai–Ramsey conjecture.

Dean conjectured that for each integer $k \ge 3$, every graph with minimum degree at least $k$ has a cycle whose length is divisible by $k$; this conjecture is known to be true for all $k\neq 5$. For $k\in\{3,4\}$, stronger statements are true: every graph with minimum degree at least $2$ and at most $k-2$ vertices of degree $2$ has a cycle whose length is divisible by $k$. We further strengthen these results by characterizing all graphs with minimum degree at least $2$ and at most three vertices of degree $2$ that have no cycle of length divisible by $k$, for each $k\in\{3,4\}$. As a corollary, we obtain that every graph with minimum degree at least $2$ and at most two vertices of degree $2$ has a cycle whose length is divisible by $3$, and that every graph on at least nine vertices with minimum degree at least $2$ and at most three vertices of degree $2$ has a cycle whose length is divisible by $4$.

In this paper, we completely classify the isomorphism classes of certain lattices $L_A(C)$ and $L_B(C)$ from a self-orthogonal code $C$ over the finite field $\mathbb{F}_p$, where $p$ is an odd prime. These lattices are obtained by Construction A and B for a code $C$ over $\mathbb{F}_p$ introduced by Lam and Shimakura, which arose from a study of orbifolds of lattice vertex operator algebras. For self-orthogonal codes $C$ and $D$ of the same length over $\mathbb{F}_p$, we show that $L_X(C) \cong L_X(C)$ as lattices if and only if $C \cong D$ as codes, where $X=A$ or $B$. This can be expected to be lattice analogues of classifications of the isomorphism classes of lattice vertex operator algebras and its orbifolds. To prove the result, we generalize the notion of a frame of a lattice and define some codes which are analogues of codes constructed from Kleinian codes studied by Höhn.

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which asserts that the square-root polynomial of the invariant polynomial $\Phi_p(G;x) \in \mathbb{F}_p[x]$ can replace its square-free part to yield a more effective criterion for a graph to be determined by its generalized spectrum (DGS). A key ingredient of our proof is a novel algebraic factorization: we show that the polynomial $\Phi_p(G;x)$ is the product of the characteristic polynomials of the adjacency operator restricted to the left null space of the walk matrix and its radical, respectively. Based on this refined DGS-criterion, a broad family of DGS-graphs is constructed via rooted products, significantly generalizing the recent result of Wang, Shen, and Mao (Discrete Appl. Math., 2026).

We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems.

Let $\zeta$ be Euclidean norm of the degree sequence of a graph normalized by the graph size. We prove that when the vertices of a graph are randomly colored with $s$ colors such that the fraction of vertices in each color class is bounded away from zero, only two asymptotic regimes emerge. If $\zeta=o(1)$, then the sizes of the subgraphs induced by the color classes concentrate around their expected values. If $\zeta=\Theta(1)$, then concentration depends on the color balance: for colorings with persisting imbalance, the total number $M$ of monochromatic edges stays bounded away from its mean with positive probability; otherwise, for vanishing imbalance, $M$ still concentrates. The same dichotomy holds for a broad class of randomly colored random graphs.

This paper generalizes in the affine symmetric group the notion of Coxeter sortable (or c-sortable for short) elements, as well as the classical bijection between c-sortable elements and c-noncrossing partitions defined by Reading in finite Coxeter groups. The generalization to the affine symmetric group of the c-sortable elements is achieved by using biclosed sets of reflections. Using recent works from Barkley and Speyer, these biclosed sets admit a sort of "one-line notation" called a TITO on $\mathbb{Z}$ (translation-invariant total order on $\mathbb{Z}$) that coincides with the usual one-line notation in the case of an affine permutation. We characterize the c-sortable elements of the affine symmetric group by pattern avoidance on their one-line notation, mirroring the well-known characterizations of c-sortable elements in the classical finite types. Based on this criterion, we then define the c-sortable biclosed sets, generalizing the c-sortable elements, as biclosed sets such that their TITOs on $\mathbb{Z}$ avoid certain patterns. We also build a bijection from our set of c-sortable biclosed sets to the set of c-noncrossing partitions using various combinatorial objects and their one-to-one correspondences. First, the TITOs, in bijection with the biclosed sets of the affine symmetric group using results from Barkley and Speyer. Second, the c-noncrossing partitions of an annulus, in bijection with the c-noncrossing partitions of the affine symmetric group, using results from Digne and Reading. Finally, the cyclic noncrossing arc diagrams, defined by Barkley, for which we exhibit a subset in bijection with both the set of c-sortable biclosed sets and the set of c-noncrossing partitions.

The book number $b(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. A strengthening of Mantel's theorem due to Rademacher states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor$ edges contains at least $\lfloor n/2\rfloor$ triangles. Another strengthening, initiated by Erdős, asserts that every such graph $G$ satisfies $b(G)\ge n/6$. Motivated by these results, Mubayi studied the tradeoff between the total number of triangles and the book number in such graphs, and asymptotically resolved the problem when $n/4\le b(G)\le n/2$. Conlon, Fox, and Sudakov conjectured that, for $n/6\le b< n/4$, every $n$-vertex graph with at least $\lfloor n^2/4\rfloor$ edges and book number at most $b$, other than the balanced complete bipartite graph, has at least $b^2(n-4b)$ triangles, with equality only for the blow-up $S_{b,n}$ of the $3$-prism. They proved the conjecture when $b$ lies in an interval with endpoint $n/4$, and also at the endpoint $b=n/6$, where they asked whether it remains valid in an interval containing this endpoint. In this paper, we answer this question affirmatively. We show that there exists a constant $\varepsilon>0$ such that the conjecture holds for all $n/6\le b\le (1/6+\varepsilon)n$. Our proof first establishes a stability theorem showing that every extremal graph is close to a blow-up of the $3$-prism, and then uses a detailed parameter analysis to force the exact six-partite structure.

The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we investigate structural conditions ensuring that tensor factorizations and generalized inverses admit real realizations. We show that these conditions can be characterized through the conjugate-pairing structure of the Fourier frontal slices, which determines when transform-domain constructions yield real tensors after inverse transformation. As applications, we obtain real versions of several tensor factorizations and analyze the existence and structure of associated generalized inverses. These results provide a framework for transferring matrix-based constructions to real tensors while preserving the algebraic constraints of the $t$-product.

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal–Katona theorem implies that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_\ell\mathcal{F}|$ be? We call a hypergraph extremal if it achieves the minimum value of $|\partial_\ell \mathcal{F}|$ subject to the degree condition $\delta(\mathcal{F}) \geq \binom{t}{k-1}$. Füredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] proved that for $k=3$, $\ell=2$ and $t\ge 2$, every extremal hypergraph contains an isolated copy of $K_{t+1}^3$ when $|X| > \frac{1}{4}(t+1)^2(t+2)$. In this article, we study the general case $k > \ell \geq 2$. By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that, for every integer $t\ge k-1$, there exists an extremal hypergraph containing an isolated copy of $K^{k}_{t+1}$ whenever $|X| > \frac{1}{4}(t+1)^2\binom{t-1}{\ell-2} + 3t+1$. In the case when $k=3$ and $\ell=2$, this gives the threshold $\frac14(t+1)^2+3t+1$, which is smaller than $\frac14(t+1)^2(t+2)$ for every $t\ge3$; for $t=2$, the two thresholds give the same integer condition on $|X|$.

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_{k-1}\mathcal{F}|$ be? In this article, for the case $k=3$, we prove that, for every sufficiently large integer \(t\), every extremal hypergraph for this problem contains an isolated copy of $K_{t+1}^3$ whenever $|X| \geq ct^2 + o(t^2)$, with the constant $c = 1 + \sqrt{928/33}$. Our proof uses a graph transformation that regularizes the neighborhood structure of extremal graphs, reducing the problem to a counting argument on the neighbors of a disjoint clique family. This gives a quadratic-order threshold for the every-extremal version of the problem, compared with the cubic-order threshold of Füredi and Zhao [SIAM J.\ Discrete Math.\ 36(4), 2022].

For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. We prove that \[ g(k,m)\le \max(m+2k-2,\,3k+1) \] for all $k\ge 1$ and $m\ge 2$, establishing the 1983 conjecture of Thomassen that $g(k,k+1)\le 3k+1$. The key new ingredient is a Hall-feasibility argument replacing the final numerical step in the proof of Nguyen.

In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi's MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford–Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford–Weil groups and span the invariant subspaces of these groups.

We study conic divisorial ideals from the viewpoint of matroid theory and apply the resulting framework to toric rings arising from root systems. For a toric ring, we describe the polytope representing divisor classes corresponding to conic divisorial ideals in terms of matroids. We then turn to the toric ring $R_P$ associated with a certain subset $P$ of a classical root system, called a signed poset. We compute the divisor class group and characterize the ($\mathbb{Q}$-)Gorenstein property of $R_P$ in terms of $P$. Moreover, we also construct a polytope characterizing the conic divisorial ideals of $R_P$. This recovers and extends previous results on Hibi rings to our toric rings.

In this work, we explore the interplay between graph limit theory, the geometry of underlying probability spaces, spectral theory, and network dynamical systems. We investigate two primary questions concerning forward and inverse perspectives: first, whether a graphon retains information about the geometry of the space on which it is defined, and second, whether spectral properties can distinguish graphons that originate from different geometric spaces. To address these questions, we differentiate between combinatorial equivalence and geometric structure, highlighting how these concepts are captured simultaneously by the class of pure graphons. Furthermore, we construct explicit examples of isospectral graphons – graphons whose integral operators share the same spectrum – that differ in their underlying geometry. By utilizing the heat kernels of Neumann- and Dirichlet-isospectral drums, we demonstrate that these graphons are not combinatorially equivalent. Finally, we establish new connections between the geometric aspects of graph limit theory and dynamical systems by analyzing a continuum Kuramoto model with graphon-defined interactions. We demonstrate that while isospectrality implies identical stability properties in certain cases, this correspondence breaks down when the differing boundary conditions of our specific Neumann and Dirichlet constructions are considered.

We study the volume map on Artinian quotients of Cohen-Macaulay algebras in characteristic $p$, and the interaction between it and the action of Frobenius on resolutions. This allows us to provide a general, conceptual way to understand Parseval-Rayleigh identities, curious inhomogeneous identities on the volume map which were developed for the proof of the Ohsugi-Hibi conjecture. This general perspective gives a new approach to generic Lefschetz theory. We use this perspective to do the following: we give sufficient conditions for anisotropy and the Hard Lefschetz property for generic Artinian reductions of graded Gorenstein rings; we study the codimension-$3$ Gorenstein quotient of a polynomial ring by the ideal generated by Pfaffians, proving a Parseval-Rayleigh identity and deriving anisotropy and Hard Lefschetz in characteristic $2$; we deduce the $g$-theorem for simplicial spheres and the Ohsugi-Hibi conjecture following previous work of Adiprasito, Papadakis, and Petrotou; and we provide further examples of Parseval-Rayleigh identities for Gorenstein rings.

An inclusion of nonvoid simplicial complexes induces an arrow of polyhedral products. Applying Ayzenberg's polyhedral join to both complexes gives a symmetric relative join operad. Its endpoint suboperads recover Abramyan–Panov substitution and Ayzenberg composition. Neither endpoint operad nor either relative join operad is finitely generated. The suboperad with nonvoid lower complex arises as the nonempty power-set quotient of the subset-inclusion operad. When tensoring preserves colimits of nonempty finite diagrams, the suboperad acts on arrows by polyhedral colimits up to coherent natural isomorphism. The action induces a set-operad algebra on isomorphism classes of arrows. The dual construction produces Stanley–Reisner quotient arrows, and the colimit action refines Eldridge's loop-space decomposition to arrows. PL ball–boundary pairs form a suboperad of the relative join operad, and minimal interior faces give an operad morphism. Principal pairs have odd-dimensional spherical moment-angle homotopy fibers and yield a closure result for Eldridge's loop-space class.

In this paper, we provide a complete structural characterization of which words in noncommuting matrix variables and their formal transposes universally evaluate to matrices with nonnegative eigenvalues or nonnegative trace, regardless of the dimension or entries of the substituted matrices. Crucially, we establish that in this pure monomial setting, both spectral and trace positivity exhibit an exact rigidity phenomenon analogous to Hermitian-square representations. This contrasts sharply with the broader polynomial setting, where tracial rigidity is known to fail even for approximation. Furthermore, we establish that this structural rigidity extends seamlessly to the complex and infinite-dimensional domains. We show that the exact same purely combinatorial conditions completely characterize matrix words evaluated over complex matrices, bounded operators on arbitrary Hilbert spaces, and positive tracial states on $C^*$-algebras and von Neumann algebras. As one application of our main theorems, we fully resolve open questions initially posed by Lieb and Pierce, and later conjectured by Hillar and Johnson, concerning the underlying algebraic structure of the summands in Bessis-Moussa-Villani (BMV) trace polynomials from quantum statistical mechanics. We achieve these results by establishing a surprising connection to graph theory, leading to a matrix analogue of the positive graph conjecture posed by Camarena, Csóka, Hubai, Lippner, and Lovász.

A graph $G$ on $m$ edges is graceful if there is an injection $f : V(G) \to \{0, 1, \ldots, m\}$ whose induced edge labels $\{|f(u) - f(v)| : uv \in E(G)\}$ are exactly $\{1, 2, \ldots, m\}$. Ringel and Kotzig conjectured in 1964 that every tree is graceful. A computer check has confirmed this for all trees on at most 35 vertices (Fang 2010), but no general proof is known. Here we exhibit an infinite family of trees that escapes the named spider results of Bahls–Lake–Wertheim, Panpa–Poomsa-ard, and Panpa–Imnang–Wasuanankul: the family $T(k, m)$ of spiders with $k$ legs of length $3$ together with $m$ pendant leaves at the centre. We prove every such tree is graceful for all $k \ge 1$ and $m \ge 0$. The argument splits into two short lemmas. The first is a pendant-extension lemma that applies whenever the underlying graceful labeling sends the centre to $0$; the second is the base case, namely that $S_{k, 3}$ admits exactly such an apex-zero labeling, a fact already implicit in Bahls–Lake–Wertheim (2010). What is new is the explicit identification of $T(k, m)$ and the observation that the family is closed under both pendant addition and the apex-zero condition, so it includes infinitely many trees not handled by the named theorems above.

We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for permutons supported by graphs of measure-preserving functions, and demonstrate how the local regularity of this function obstructs approximability. We also consider the biased Brownian separable permuton and prove a lower bound on its approximation error by showing that its supporting measure-preserving function has Lipschitz points almost surely.

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\dots,x_k)$ with $1\le x_1<x_2<\cdots<x_k\le n$, where two vertices $(x_1,\dots,x_k)$ and $(y_1,\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\le i\le k-1$ or $y_{i+1}=x_i$ for all $1\le i\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd girth. We further observe that shift graphs arise naturally as induced subgraphs of simplified de Bruijn graphs. Although simplified de Bruijn graphs contain non-word-representable members in general, we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable. As a consequence, we obtain an explicit family of graphs exhibiting a contrast between line graph and line digraph constructions: there exists a family of word-representable graphs whose line graphs are not word-representable when the number of vertices is at least $5$, while their line digraphs are word-representable.

The paper investigates perfect state transfer (PST) in Grover walks on Cayley graphs over the dihedral group $D_n$. The Grover walk is a discrete-time quantum walk widely studied in quantum information processing. A Cayley graph $\operatorname{Cay}(\Gamma,S)$ is called normal if $S$ is the union of some conjugacy classes of the group $\Gamma$; otherwise, it is called non-normal. Most existing studies have been restricted to Cayley graphs over abelian groups. In contrast, we investigate both normal and non-normal cases for Cayley graphs over the non-abelian group $D_n$. By examining the parity of $n$ and the normality of the Cayley graph, we obtain a complete characterization of PST on $\operatorname{Cay}(D_n,S)$. In particular, we establish necessary and sufficient conditions for the occurrence of PST in all possible cases, and prove that PST does not occur for normal Cayley graphs when $n$ is odd. Furthermore, we construct several infinite families of normal and non-normal Cayley graphs $\operatorname{Cay}(D_n,S)$ that exhibit PST, illustrating the application of the main result. Our approach is based on the representation theory of the dihedral group.

We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression independently conjectured by Andrews et al. and Kim et al. for the same Macdonald index.

We give new examples of graphs and trees with dominating set sequences that are not log-concave. These examples were generated by PatternBoost, a transformer-based reinforcement learning software developed by Charton-Ellenberg-Wagner-Williamson. We also show: for any positive integer $m$, there exists a tree whose dominating set sequence is not log-concave for at least $m$ indices by modifying a similar construction of Bautista-Ramos for the independent set sequence. We show that a large class of caterpillar graphs has log-concave dominating set sequences. A continuous analogue of the sequence is also log-concave for all graphs.

Nikiforov's well-known spectral Turán inequality for walks states that, for every graph $G$ with clique number $\omega(G)$, $\lambda^r(G)\le w_r(G)(1-1/\omega(G))$, where $\lambda(G)$ is the largest eigenvalue of the adjacency matrix of $G$, and $w_r(G)$ is the number of walks with $r$ vertices in $G$. For $r=1$, this is Wilf's inequality; for $r=2$, it gives Nikiforov's spectral Turán theorem. Recently, Liu and Ning proved local versions of these two inequalities, strengthening both Wilf's inequality and Nikiforov's spectral Turán theorem. It is natural to ask whether Nikiforov's spectral Turán inequality for walks also admits a local strengthening. Motivated by this question, Kannan, Kumar, and Pragada conjectured the vertex-local bound $\lambda^r(G)\le \sum_{v\in V(G)} w_r(v)(1-1/c_G(v))$, where $w_r(v)$ denotes the number of walks with $r$ vertices starting at $v$, and $c_G(v)$ is the maximum order of a clique containing $v$. This conjecture is important because it gives the most natural local form of Nikiforov's spectral Turán inequality for walks. In this paper, we confirm this conjecture. More precisely, for $r\ge 2$, we prove the stronger edge-local inequality $$\lambda^r(G) \le \sum_{uv\in E(G)} \frac{c_G(uv)-1}{c_G(uv)} \bigl(w_{r-1}(u)+w_{r-1}(v)\bigr),$$ where $c_G(uv)$ is the maximum order of a clique containing the edge $uv$. Our result implies Nikiforov's spectral Turán inequality for walks and unifies several local spectral extremal results of Liu and Ning. We also determine all extremal graphs for both the edge-local and vertex-local inequalities. The main new ingredient is a Markov-chain estimate whose transition matrix is constructed from a Perron vector of $A(G)$; this estimate carries the local edge coefficient through walks of arbitrary length.

Latin squares are $n\times n$ matrices containing $n$ symbols, where each symbol appears exactly once in each row and column. They were studied by Euler, later popularized through Sudoku, and remain a rich source of difficult combinatorial search problems. Two Latin squares are orthogonal mates if, when overlaid, no ordered pair of symbols repeats. Pairs of orthogonal Latin squares exist for every order except 2 and 6, but finding orthogonal Latin squares computationally can be challenging. Satisfiability (SAT) solvers are strong at combinatorial search and have been used to resolve a number of various kinds of orthogonal Latin square problems. On the other hand, SAT solvers lack domain knowledge about Latin squares, such as the Euler-Parker algorithm for orthogonal mate construction. In this paper, we propose a hybrid method combining a SAT solver with the Euler-Parker algorithm (implemented using a Diophantine system solver) and show that the resulting solver is effective at finding certain kinds of orthogonal Latin squares. For example, certain pairs of $10\times10$ orthogonal Latin squares whose existence was unknown for over 25 years were recently found by Bright, Keita, and Stevens using a SAT solver. The hardest cases could not be solved by the SAT solver CaDiCaL within seven days, but CaDiCaL augmented with an external Euler-Parker algorithm solves these cases in a median of around 5,100 seconds.

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting r-tuples. We prove an exact dominant asymptotic for the number of homomorphisms from the free abelian group of rank r to G. The exponential base is the maximum order of an abelian subgroup of G, and the leading coefficient is the number of abelian subgroups of that order. As a consequence, the r-th root of the higher commuting probability tends to this maximum abelian-subgroup order divided by the order of G, while the r-th root of the orbit count tends to the maximum abelian-subgroup order itself. We also prove that the associated rank-generating series is rational and has a finite Dirichlet-spectrum expansion supported on abelian subgroup indices. This spectrum yields a finite linear recurrence, a finite-rank Hankel matrix, and an inverse finite-spectrum theorem: the tail of the hierarchy determines the full abelian-index spectrum. For split abelian extensions, we express the dominant base through fixed-subgroup geometry, and for abelian acting quotients, we obtain an exact subgroup-lattice formula. In the cyclic and coprime cases, this gives closed formulas for all spectral coefficients.

We prove an $\widetilde O(n^2)$ bound for the relaxation time and the log-Sobolev time (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex $(n+2)$-gon, implying a mixing time of $\widetilde O(n^2)$. The previous state of the art for the mixing time of this chain, due to Eppstein and Frishberg, was $\widetilde O(n^3)$, while the best known lower bound on the mixing time, due to Molloy, Reed, and Steiger, is $\Omega(n^{3/2})$. Our relaxation time bound makes significant progress towards Aldous' conjectured bound of $\Theta(n^{3/2})$ for the relaxation time. We improve upon the analysis of Eppstein and Frishberg by further developing the framework of transport flows introduced in the work of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.

For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $\rho = \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{\rho+\varepsilon}$ for every $\varepsilon > 0$.

Let $H$ be a finite simple graph. A one-particle continuous-time quantum walk on its line graph $\ell H$ has one mode for each edge of $H$. This paper is a corrected and reorganized account of that setting. We first record the matrix encoding of an edge-amplitude vector, its Schur state. The encoding is necessarily complex symmetric if it is to be complex linear and to retain the global phase, and it is normalized by $1/\sqrt{2}$. It is unitary for the Frobenius inner product and is closed under a canonical tensor lift to the categorical product of graphs. The tensor lift also pins down the convention. Among all unimodular phase variants the symmetric choice is the only one that survives, and every relaxed variant reduces to a sign. We then replace the labelled tensor-power description by the bosonic and fermionic Fock spaces over $\ell^2(EH)$, and assemble the standard free-particle apparatus in the form used afterwards. This covers the occupation-number basis, the second-quantized line-graph Hamiltonian, permanent and determinant formulae for many-particle transition amplitudes, and the one-particle reduced density matrix. The one graph-theoretic input is the even-Eulerian criterion, which is the zero-sum $2$-flow criterion of Wang and Hu, recalled here with a short proof to fix conventions. Its translation through the incidence identity gives a real equimodular full-support $-2$ eigenmode of $A(\ell H)$, and hence bosonic condensates of energy $-2N$ that are uniform over the edges of $H$. The fermionic case is stated separately so that the Pauli constraint is explicit.