Papers

A fundamental identity for the number of vacillating tableaux was originally obtained through the representation theory of partition algebras. We extend this identity to arbitrary differential posets and show that it, together with several analogous identities, follows directly from the structural properties of differential posets. Specializing to Young's lattice and its Cartesian powers, we further obtain new bijective proofs via a simple deletion–insertion process.

The Vietoris-Rips complex of a metric space is the simplicial complex whose faces are the subsets of points with pairwise distance bounded above by a given scale $r$. In this paper, we study Vietoris-Rips complexes on the vertex set of the $n$-dimensional hypercube equipped with the Hamming distance. These complexes are stable under the action of the automorphism group of the hypercube graph, also known as the hyperoctahedral group, which therefore acts on their homology groups. Our results completely describe the decomposition of these homology groups into irreducible representations of the hyperoctahedral group at scales $r\leqslant 3$ and $r=n-1$.

For each positive integer $n$, let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency defined by an elementary transfer of one unit between parts, followed by reordering. Let $K_n=\mathrm{Cl}(G_n)$ be its clique complex. Previous work revealed two apparently contrasting features of this family. Locally, the graphs $G_n$ become increasingly rich: local clique dimensions grow, degree landscapes refine, support jumps and simplex layers proliferate, and axial, rear, central, shell, and directional structures become more pronounced. Globally, however, the clique complexes remain homotopically simple: $K_n\simeq \bigvee^{b_n}S^2$, where $b_n=\chi(K_n)-1$. This paper gives a conceptual synthesis explaining why these facts are compatible. The local side is governed by ordered local transfer types, which determine local neighborhood graphs, degrees, local clique numbers, and local simplex dimensions. The global side is governed not by the largest local simplices, but by the overlap pattern of canonical full star/top simplices. These simplices form a good cover $\mathcal C_n$, giving $K_n\simeq N(\mathcal C_n)=N_n$. The nerve admits an anchor-cover and intersection-poset reduction $N_n\simeq \Delta(J_n)$, with $\dim\Delta(J_n)\le 2$. Thus high-dimensional local simplices occur inside contractible containers, while global topology is controlled by a low-dimensional overlap poset. Within this family, the qualitative topological problem therefore collapses to the numerical computation and interpretation of a single integer, the Euler characteristic $\chi(K_n)$, equivalently the bouquet rank $b_n=\chi(K_n)-1$.

Let $G$ be a graph, and $a$ and $b$ be integers. Suppose there are infinite number of stairs, numbering level $0$, level $1$, etc. How do you place every vertex of the graph $G$ on a level as high as possible such that a vertex placed on level $i$ should have at least $i-b$ neighbors among the vertices placed on level $i$ and above while have at least $i$ neighbors among the vertices placed on level $i-a$ and above? This new game on graphs is referred to as “network pearls climb stairs". In this paper, we develop a more general theory, notably a correspondence between structure and dynamics, which particularly leads to the optimal solution of the game. Moreover, as $a$ and $b$ vary, the corresponding level numbers for a vertex obviously provide a structural profile (or spectrum) for the vertex the applications of which are worthy of further study.

The minimum positive codegree $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The positive codegree Turán density of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

We prove that every graph admits a linked, componental, rooted tree-cut decomposition of finite adhesion that displays all undominated edge-ends. As a first application, we deduce that this tree-cut decomposition also displays the edge-degrees of all undominated edge-ends. For locally finite graphs $-$ where every end is an undominated edge-end $-$ this yields a linked tree-cut decomposition of finite adhesion into $\textit{finite}$ parts that displays all ends and their edge-degrees. As a second application, this latter tree-cut decomposition yields short, unified deductions of Thomassen's theorem on boundary-linked finite partitions, and of Bruhn and Stein's characterisation of Eulerian locally finite graphs in terms of even ends.

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

We say that a convex polytope has the clique-face property if every clique in its 1-skeleton is the vertex set of a face. We establish this property as a geometric necessary condition for quadratic generation of toric ideals. More precisely, we prove that every lattice polytope with primitive edges and a quadratic toric ideal has the clique-face property; in particular, this holds for every $(0,1)$-polytope with a quadratic toric ideal. For $(0,1)$-polytopes satisfying condition (E), we characterize the clique-face property in terms of divisibility by monomials occurring in quadratic binomials, and show that, under the clique-face property, such toric ideals have no indispensable monomials of degree at least three. For edge polytopes and cut polytopes, we prove that the clique-face property is equivalent to quadratic generation. This yields new geometric characterizations of quadratic generation for these classes. We also prove that all simple polytopes, matroid independence polytopes, and matroid base polytopes have the clique-face property, and discuss the case of stable set polytopes in connection with conjectures on quadratic toric ideals.

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. We distinguish classical Ziegler pairs, for which the minimal degree of a Jacobian relation denoted by ${\rm mdr}$ differs, from the more general notion detected by distinct graded Betti data. First, we show that for arrangements of $d<9$ lines the intersection lattice determines ${\rm mdr}$, and hence classical Ziegler pairs do not occur in this range. Then we list several distinct Ziegler pairs with $d=10$ in the broader sense. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions. Another rather surprising consequence is that being maximal Tjurina for a line arrangement is not determined by the combinatorics.

We define a combinatorial property of a projective Fraisse category which we call the approximate Ramsey property. Let $F$ be a continuum, $G$ a closed subgroup of the homeomorphism group of $F$, and $\mathbb{F}$ the limit of projective Fraisse category $\mathcal{F}$ such that $\textrm{Aut}(\mathbb{F})$ is dense in $G$. We prove that $\mathcal{F}$ has the approximate Ramsey property if and only if $G$ is extremely amenable. We prove that the group of homeomorphisms of the universal pseudo-solenoid has non-metrizable universal minimal flow.

We prove that every $K_5$-free $n$-vertex graph with sublinear independence number can be made bipartite by removing at most $n^2(1/18+o(1))$ edges, where the constant $1/18$ is best possible. The proof method is related to extensions of Turán Theorem in edge-weighted settings, and part of the proof uses flag algebra.

In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $\Omega(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.

$q$-Matroids, a $q$-analogue of classical matroids have attracted a lot of attention over the last decade, yet their enumeration remains largely unexplored. In this paper, we study the number of $q$-matroids, paving and sparse-paving $q$-matroids defined on a fixed ground space and with prescribed rank. We derive new lower bounds using constructions from constant-dimension codes and improve existing estimates. On the upper bound side, we develop two approaches: a combinatorial method based on controlling the number of dependent hyperplanes for paving $q$-matroids, and an entropy-based counting argument applicable to classes of $q$-matroids closed under contraction. These techniques yield explicit upper bounds on the logarithmic number of $q$-matroids with fixed rank and ground space. Finally, we analyze the asymptotic behavior of these bounds, and identify gaps between lower and upper estimates, leading to conjectures on the true asymptotic growth.

We continue recent work of Chao, Gabel, Larson, and Nasr in using cyclotomic polynomials for dice relabeling. In their work, one idea they expand on is finding pairs of dice with different number of sides which maintain the sum frequency of two normal dice. We continue this idea in this paper by studying pairs of dice where the number of sides of each is a different perfect square (which we call "square-sided" dice). We additionally provide conjectures offering ideas for future exploration.

The toughness of a graph is a crucial parameter for characterizing its structural properties. The toughness of a non-complete graph $G$ is defined as $\tau(G) = \min \{ \dfrac{|S|}{c(G - S)} : S \subseteq V(G), c(G-S) > 1 \}$, where $c(G)$ denotes the number of components of $G$. We define $\tau(K_n) = \infty$. A graph $G$ is said to be $\tau$-tough if $|S| \ge \tau \cdot c(G-S)$ for every vertex cut $S$ of $G$. Let $k \ge 3$ be an integer. For $\frac{1}{k-\eta}$-tough graphs with $\eta \in \{0, 1\}$, Liu, Fan and Shu \cite{a34} derived sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Jia and Lu \cite{a24}, for the case $\frac{1}{k-1} \leq \tau(G) < \frac{1}{k-2}$, established sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Motivated by these results, in this paper, we further investigate sufficient conditions for the existence of a spanning $k$-tree when $\frac{1}{k} \leq \tau(G) < \frac{1}{k-1}$. Specifically, for a connected $\frac{t}{t(k-1)+1}$-tough graph of sufficiently large order $n$ (where $t \ge 1$ is an integer), we provide sufficient conditions for the existence of a spanning $k$-tree in terms of the spectral radius and the signless Laplacian spectral radius. Furthermore, we establish a lower bound on the size (number of edges) to guarantee the existence of a spanning $k$-tree.

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

We study properties of graphs (or rather graph sequences) saying that some restricted count of subgraphs is approximatively what is expected in a random graph. It has been shown by several authors that many such properties characterize quasi-random graphs, but there are also some exceptions. We continue here the line of investigation in Janson and Sós (2013), and introduce some new versions of these properties, in order to better understand why many of these properties are quasi-random, and to understand the structure of the exceptions that are not. A new feature in the proofs is a simple decomposition of the subspace of symmetric functions in $L^2([0,1]^m)$ into subspaces that are irreducible for the action of measure-preserving transformations of $[0,1]$; this simplifies some arguments and gives structure to others.

The study of intersecting families, initiated by Erdős, Ko, and Rado, is a central topic in extremal combinatorics. A classical stability result of Hilton and Milner determines the largest non-trivial intersecting family, and in subsequent works researchers developed structural stability results via the notion of diversity. In this paper, we study cross-intersecting families. We establish a structural theorem for large cross-intersecting pairs, extending Kupavskii's theorem from intersecting families to the cross-intersecting setting. Our result characterizes extremal cross-intersecting pairs in terms of their diversity parts and maximal cross-intersecting extensions. As corollaries, we obtain cross-intersecting analogues of several classical theorems, including those of Han–Kohayakawa and Huang–Peng. A key ingredient in the proof is a new shifting method, called the $S_{U,V}^{Q}$-shift, which not only preserves global intersection properties but also maintains certain local substructures after shifting. We expect this method to be useful elsewhere, and it is already one of the key tools in establishing a product analogue of the Hilton–Milner theorem.

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

In the regular three-eigenvalue setting, spectral complexity and coherent-algebraic complexity coincide: a connected regular graph has exactly three distinct eigenvalues if and only if it is strongly regular, its coherent rank is three. Although examples of regular graphs with four distinct eigenvalues and coherent rank larger than four are known, it was unknown whether coherent rank is uniformly bounded among regular graphs with four distinct eigenvalues. We show that no such bound exists, even under the additional assumption of co-edge-regularity. For every prime power \(q\), we construct infinitely many co-edge-regular graphs with exactly four distinct eigenvalues, smallest eigenvalue \(-2q-1\), and coherent rank at least \(q+4\). Consequently, coherent rank is unbounded among co-edge-regular graphs with exactly four distinct eigenvalues.

Keevash and Michaeli recently proved that, under the robustness assumption that \(G\) is an \(s\)-connector (i.e. \(\overline G\) is \(K_{s,s}\)-free), \(G\) has essentially the same multicolour Ramsey matching properties as complete graphs, with an additive error \(O(qs)\), where \(q\) is the number of colours. They asked whether the dependence on \(q\) can be removed. We answer this question in a stronger exact form. For \({\bf t}=(t_1,\ldots,t_q)\in\mathbb N_+^q\), let \(R_s({\bf t})\) be the smallest integer \(N\) such that every \(N\)-vertex \(s\)-connector \(G\) satisfies \( G\to (t_1K_2,\ldots,t_qK_2). \) We determine the exact value \[ R_s({\bf t})=\sum_{j\in[q]}(t_j-1)+ \max\left\{2s,\ s+\max_{j\in[q]}t_j\right\}. \] While Keevash and Michaeli's proof uses a compression algorithm based on the Gallai–Edmonds decomposition to reduce the colouring to a structured form, our proof is a direct minimal-counterexample argument together with a new counting method for monochromatic matchings which can be applied to \(s\)-connectors.

The clique adjacency polynomial (CAP), introduced by Soicher (2015), provides a powerful method for bounding the clique numbers of edge-regular graphs. In this paper, we extend the CAP framework to arbitrary finite graphs by expressing the relevant parameters in terms of average vertex degree and average edge-degree over potential cliques. This leads to a generalised CAP bound and an associated clique existence polynomial (CEP), which removes the dependence on an auxiliary integer variable and facilitates computation. We compare the resulting bounds with classical spectral and linear programming bounds, including those of Delsarte, Hoffman, and Haemers. We show that the generalised CAP improves upon these bounds for several families of graphs. In particular, we identify infinite families of edge-regular graphs arising from projective geometry for which the CAP outperforms the Delsarte bound, as well as families of regular and non-regular graphs where the generalised CAP improves upon the Hoffman and Haemers bounds. We also develop techniques for bounding feasible parameter regions, enabling practical application of the method to both structured and unstructured graphs.

Introduced by Ballantine, Beck, and Merca, the elementary symmetric partition function $\mathrm{pre}_k$, defined on the set of partitions with at least $k$ parts, has been a topic of recent interest. We prove that $\mathrm{pre}_k$ is injective on the set of $m$-ary partitions for positive integers $m \ge k$, generalizing the binary $k = 2$ result of Ballantine, Beck, and Merca, and complementing a result of Hadelyn, Niergarth, Li and Li showing that, for each $k \ge 3$, $\mathrm{pre}_k$ is not injective on partitions of $n$ with length $2k$ for infinitely many $n$. We introduce the skew Schur partition function $\mathrm{prs}_{\lambda'/\mu'}$, prove injectivity results for particular choices of $\lambda',\mu'$, and describe an application to representation theory.

For a finite simplicial complex on $[n]$, the combinatorial Hodge Laplacian splits as $L_k=L_k^{\mathrm{up}}+L_k^{\mathrm{down}}$, and Duval and Reiner showed that $\lambda_{\max}(L_k^{\mathrm{up}})\le n$ in every dimension. We conjecture that $\lambda_{\max}(L_k^{\mathrm{up}})$ is in fact non-increasing in $k$, equivalently that $\sigma_{\max}(\partial_{k+1})\le\sigma_{\max}(\partial_k)$, and prove this unconditionally in two cases: when every missing $(k+1)$-face has at most $k+1$ missing facets, and for shifted complexes, where we also identify the extremal eigenvalue exactly, as the number of vertices lying in a $(k+1)$-face. In general we prove \[ \lambda_{\max}\big(L_k^{\mathrm{up}}\big)\ \le\ \nu_{k-1}+\tfrac1{k+2}\big(n-\nu_{k-1}\big), \qquad \nu_{k-1}=\lambda_{\max}\big(L_{k-1}^{\mathrm{up}}\big), \] refining that ceiling. The proofs run through a localization on the cycle space $\ker\partial_k$, which turns the comparison into a statement about the complement. In dimension one the complex is the clique complex of a graph, $L_1$ is its Helmholtzian, the conjecture is a question of Lu, Shi, Stanić, Wang and Wang, and the first case reads $\alpha(G)\le2$. We also characterize the connected graphs of order at least seven with $\lambda_2(L_1)\le 3$ as the firefly graphs.

For a family of graphs $\mathcal F$, a graph $G$ is said to be $\mathcal F$-free if it contains no member of $\mathcal F$ as a subgraph. A wheel graph $W_k$ is a graph on $k+1$ vertices formed by joining a new vertex to all vertices of a $k$-cycle. Given an integer $k\ge 3$, we consider the problem of determining the maximum number of triangles in a $W_{\geq k}$-free graph, where $W_{\geq k}=\{W_\ell: \ell \geq k\}$. The case $k=3$ was raised by Gallai, who proposed a conjecture for this case (see Erdős [5]. Gallai's conjecture was disproved by Zhou [17] and independently by Füredi, Goemans, and Kleitman [9]. In this paper, we study the case $k=4$. Namely, for every integer $n\ge 3$, we determine the maximum number of triangles in an $n$-vertex $W_{\geq 4}$-free graph and characterize all extremal graphs.

Let $\cpsi_{2,a}(n)$ denote the number of $(2,a)$-colored Frobenius partitions of weight $n$, where the two rows have prescribed length difference. We study the two cases $a=0$ and $a=1$ and connect them with minimal-excludant statistics on bipartitions. Let $\sigma\mex_2(n)$ be the sum of the Lin–Liu bipartition minimal excludants over all bipartitions of $n$, and let $E_2(n)$ be the number of bipartitions whose two component minimal excludants are equal. For all $n\geq 0$, we give a combinatorial proof of \[ \cpsi_{2,0}(n)=2\sigma\mex_2(n) \qquad\text{and}\qquad \cpsi_{2,1}(n)=2\sigma\mex_2(n)-E_2(n). \] These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.

We show, for example, that if one samples \[\frac{\log p}{2\log(1+(p-1)^{-1})} \cdot n^2(1 + o_{n\to \infty}(1))\] points in $\mathbb{F}_p^n$ at random then asymptotically almost surely this set intersects every quadratic hypersurface. We furthermore show that this is tight in that sampling $o_{n\to\infty}(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface. Our main result is a sharp threshold for the following problem: how many points in $\mathbb{F}_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most $s$ polynomials each of degree at most $k$? As an application we improve lower bounds in the random Szemerédi theorem in $\mathbb{F}_p^n$, in particular obtaining a leading constant which grows as the threshold for what is considered a `dense' set in Szemerédi's theorem shrinks.

Harborth's conjecture states that every planar graph has a crossing-free straight-line drawing in which every edge has an integer length. Kleber's strengthening asks for the vertices themselves to have integer coordinates. In this series of papers, we make progress towards settling these conjectures. We reduce Kleber's conjecture to local rational-distance statements for special polygons with at most five vertices. The triangle case is known from the results of Almering and Berry. In this paper, we prove the existence of a rational-distance point on an interior integer diagonal in all the cases for non-degenerate quadrilaterals. In the upcoming papers, we focus on non-degenerate pentagons and then degenerate quadrilaterals and degenerate pentagons.

In this article, for a convex lattice polytope, we further investigate the summatory function of its Ehrhart polynomial, which is called the summatory Ehrhart polynomial, and introduce its summatory Ehrhart series. We prove several fundamental properties of these invariants. In particular, we derive a summatory analogue of the classical Ehrhart–Macdonald reciprocity law, which establishes a signed functional equation between the polytope and its relative interior via the substitution $t \mapsto -t-1.$

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a graphic oriented matroid or a Lawrence oriented matroid, thus confirming the conjecture in these cases.