Papers

We show that very small distances in a planar integral point set are essentially one-dimensional. Let P be a non-collinear set of n points in the plane, all of whose pairwise distances are integers. We prove that, for a sufficiently small $c>0$, at most one pair of points can determine a distance below $n^{c\log\log{n}}$ unless all such short pairs are supported on a single line. We also give a construction showing that the line-supported alternative is necessary: there are arbitrarily large non-collinear integral point sets with one off-line point and with the first two distinct distances 2 and 4. We conjecture that in the general case (no three points are collinear) even the smallest distance should be large, at least $n^{c\log\log{n}}$, however we can prove a linear lower bound only.

We show that the polar degrees of the coordinate-wise inverse of a linear subspace $L \subseteq \mathbb{P}^n$ are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid $\mathrm{M}(L)$. Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of $\mathrm{M}(L)$. As a corollary, we settle two open conjectures regarding matroid discriminants.

Scott and Sokal asked for a structural description of the finite graphs \(G\) for which inverse powers \(T_G^{-\beta}\) of the spanning-tree polynomial are completely monotone. We prove the following bounded-treewidth criterion: if \(G\) is a finite connected simple graph with \(\operatorname{tw}(G)\le k\), then \(T_G^{-\beta}\) is completely monotone for every \(\beta>(k-1)/2\). Consequently, every partial \(3\)-tree is covered throughout the first Scott–Sokal open interval \(1<\beta<3/2\), including finite Apollonian networks, \(K_5-e\), and the four-spoke wheel \(W_4\). The proof combines the real Riesz/Wishart integral for determinants, star–mesh elimination of simplicial vertices, and a Gaussian Laplace kernel for the degree-\(d\) star. General bounded-treewidth graphs follow by chordal completion and monotone deletion of completion edges.

We investigate ideas from the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001) in the setting of graph isomorphism. This provides us a playground of finite combinatorial objects on which to explore these techniques. We seek to separate non-isomorphic pairs of graphs using vector spaces of polynomials that are set-wise invariant under permutations (so-called separating modules). We characterize the power of this method for distinguishing graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ are equivalent in power to the counts of $O(k)$-vertex subgraphs. - We show that separating modules of symmetric algebraic circuit size $n^{\Theta(k)}$ are equivalent to $\Theta(k)$-dimensional Weisfeiler-Leman. This generalizes and strengthens the result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025). - When considering only the representation-theoretic multiplicities of separating modules, we show that two graphs are separated by multiplicities if and only if their automorphism groups have different multiplicity of cycle types (cycle index). The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect support size (from the study of WL) to complexity measures on $S_n$ (Dafni, Filmus, Lifshitz, Lindzey, & Vinyals, ITCS '21); as well as connections between invariant polynomials, the Graph Reconstruction Conjectures, and Forman's "invariants of finite type" (Adv. Math., 2004).

For a set of $m$ subsets of a universe set of size $n$, we construct a Boolean polynomial $\mathrm{ICC}_{m,n}(X)$ such that the Union-Closed Conjecture is true for this $m$ and $n$ if and only if $\mathrm{ICC}_{m,n}(X)$ is the zero Boolean polynomial. We use an equivalent formulation, called the Intersection-Closed Conjecture.

For a general curve in positive characteristic, taking the Frobenius pullback induces a generically finite rational map V on the moduli space of rank 2 vector bundles with trivial determinant. Recently, Kondo–Wakabayashi show that the generic degree of V, considered as a function on the characteristic of the base field, is a quasi-polynomial. In this paper, we show that this quasi-polynomial is indeed a polynomial, and we write out this polynomial explicitly.

We efficiently conflict-free color every planar graph with 4 colors. An (open-neighborhood) conflict-free coloring assigns colors to vertices in a way that every vertex v has a neighbor w such that the color of w is distinct from the colors of the other neighbors of v (i.e., the color of w is unique in the open neighborhood of v). A previous best upper bound on the conflict-free chromatic number of planar graphs was 5, and it is known that 4 colors are sometimes necessary. Deciding whether, e.g., a planar graph admits a conflict-free coloring with 3 colors is NP-complete. Our approach uses a refined variant of the classical Gallai-Edmonds decomposition and the Four Color Theorem. In fact, our result is equivalent to the Four Color Theorem.

This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) <doi: 10.48550/arXiv.2606.23853> introduce a coherence criterion that characterizes exactly when qualitative majority judgments are representable by a finitely additive measure. The question addressed here is whether that coherence criterion can be replaced, in the finite setting, by any bounded finite fragment. We prove that it cannot. For every $k\ge 1$, we construct a maximal standard frame whose shortest coherence violation has length exactly $2k+2$. Hence there is no uniform finite bound on the incoherence index of social decision frames, resolving Conjecture 5.7 stated by Moss and Pedersen (2026). The construction is geometric, in the sense that it proceeds via orthogonality and dimension in rational vector spaces, and self-contained: it isolates a symmetric family of half-sized voting blocs and extends it to a maximal frame in which every shorter balanced obstruction is excluded. Along the explicit infinite sequence of universe sizes obtained in the construction, this also establishes the middle-layer family predicted by Conjecture B.25 by Moss and Pedersen (2026). Together with the soundness and completeness theorem for the Moss-Pedersen minimal logic for strict majorities, this establishes that measurable social decision frames are not finitely axiomatizable in that language.

In connection to the Brown-Erdős-Sós conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \ge 3$, $k \ge 3$, and suppose that $n \ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \[|E(H)| \geq \frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \frac{n}{r},\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \geq \frac{r-1}{r} \cdot \frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of the large-uniformity form of the Brown-Erdős-Sós theorem, due to Keevash and Long. In the case of triple systems, our bound becomes $c \geq \frac{2(k-2)}{3(k-1)} + o(1)$, improving upon a bound of $\frac{4}{5}$ due to Santos and Tyomkyn.

The theory of Latin squares has a long history. While the objects themselves appeared earlier, the study of their general mathematical theory dates back to Euler in the 18th century. Latin squares can be interpreted as 3-dimensional permutation hypermatrices, and alternating sign matrices often arise as a natural generalisation of permutation matrices. In 2018, Brualdi and Dahl introduced a generalisation of classical Latin squares using alternating sign hypermatrices. Inspired by their definition, we develop the theory of Italian squares, a related generalisation of Latin squares, together with a notion of orthogonality that resolves an inconsistency in the definition of Brualdi and Dahl. Building on classical questions from Latin square theory, we obtain results including upper bounds on the maximal size of a pairwise orthogonal set, conditions for the existence of an orthogonal mate, infinite families of orthogonal pairs, and transversals. As part of our exploration of alternating sign matrices, we also prove a Frobenius-König type result for a class of $(0,\pm1)$-matrices.

Let $n$ and $p$ denote the numbers of vertices and labels, respectively, in an undirected edge-labeled graph. Previous work showed that, under the Exponential Time Hypothesis (ETH), there is no deterministic algorithm with running time \[ (np)^{o\left(\frac{\log n}{(\log\log n)^2}\right)}. \] In this paper, we give a deterministic reduction that strengthens this conditional running-time lower bound to \[ (np)^{o(\log n)}\operatorname{poly}(|E|). \] The lower bound holds even for simple edge-labeled graphs. Since our reduction is deterministic, the same lower bound applies to bounded-error randomized algorithms under the randomized Exponential Time Hypothesis. On the resulting hard family, $p=n^{O(1)}$. Thus, under rETH, the lower bound matches the exponent order of the known randomized quasi-polynomial exact upper bound up to constant factors.

Nahm sums and Rogers-Ramanujan type identities have attracted considerable attention in recent years. In this paper, we provide analytic proofs of five conjectural identities on modular rank four Nahm sums that were proposed by Cao and Wang. Moreover, we reveal that the conjectures of Shi-Wang and Cao-Wang are closely related.

We analyze the decrease of simplex diameters under iterated refinement of spherical Delaunay complexes. Unlike in ordinary subdivision, the refined Delaunay complex need not be a subdivision of the previous one, so mesh contraction is not automatic. We derive explicit contraction bounds for several families of Steiner points, including Delaunay analogues of barycentric and edgewise subdivision. The proof reduces the problem to sharp covering estimates for Euclidean simplices. These estimates are obtained through a strengthening of Maurey's empirical method via pivotal sampling and a dimension-dependent version of the approximate Carathéodory theorem. Theoretical results and numerical experiments show that Delaunay refinements achieve stronger contraction than their subdivision counterparts.

Motivated by sequence reconstruction and reconstruction codes, we study extremal intersections of deletion balls over a fixed $q$-ary alphabet. Let $\Sigma_q^n$ be the set of sequences of length $n$ over $\Sigma_q$, and let $D_t(x)$ denote the set of all sequences obtained from $x\in\Sigma_q^n$ by deleting exactly $t$ symbols. Our first result gives a finite upper bound under a lower-order deletion-correction constraint. We prove that if $x,y\in\Sigma_q^n$ satisfy $D_{s-1}(x)\cap D_{s-1}(y)=\varnothing$, then \[ |D_t(x)\cap D_t(y)| \le \binom{2s}{s}\binom{n-s}{t-s}. \] For binary alphabets, this strengthens a recent asymptotic upper bound of Pham, Goyal, and Kiah (2025, JCTA). We then investigate deletion-ball intersections under simultaneous constraints on run counts and lower-order deletion-ball intersections. For fixed $0<\gamma\le1$, integers $1\le s\le t$, and $m\ge1$, we show that if $x,y\in\Sigma_q^n$ have at most $\gamma n$ runs and satisfy $|D_s(x)\cap D_s(y)|\le m$, then \[ |D_t(x)\cap D_t(y)|\le \frac{m\gamma^{t-s}}{(t-s)!}n^{t-s}+O_{s,t,m}(n^{t-s-1}). \] Moreover, the leading term can be attainable whenever $m$ is realized by a fixed finite-length seed pair. As a consequence, we obtain a direct lifting theorem for deletion reconstruction codes, transferring reconstruction properties from radius $s$ to larger radii $t$. Finally, we establish a parallel insertion theory and derive corresponding results for insertion-ball intersections and insertion reconstruction codes.

For graphs $H$ and $F$, the generalized Turán number $\operatorname{ex}(n,H,F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We prove a general lifting principle for the case $H=K_3$ and the forbidden graph is a vertex-disjoint union of several copies of a graph. The key hypothesis is a local neighborhood-forcing condition: there is a graph $R$ with $\operatorname{ex}(n,R)=o(n^2)$ such that $F\subseteq K_1\nabla R$. Under this condition, the corresponding single-forbidden-graph asymptotics, together with a construction attaining the relevant extremal triangle and edge densities simultaneously, lift to an asymptotic value for \(\operatorname{ex}(n,K_3,(s+1)F)\) for every integer \(s \). We also prove an exact version in terms of the maximum value of a weighted expression over all graphs of a given size that avoid the forbidden graph. As applications, we obtain exact or asymptotic results for disjoint unions of suspensions of paths and stars. We also recover known exact results for disjoint odd cycles.

In this paper, we investigate a set of $k$-color partitions and examine how its subsets relates to the theory of the minimal excludant function introduced by Andrews and Newman. Furthermore, we provide generalization of the partition functions introduced by Andrews and Bachraoui and also define several new partition functions within this framework.

In 1996, Youngs proved a surprising theorem that quadrangulations of the projective plane could never have chromatic number exactly 3. This sparked a lot of interest, and the result has been further developed in many directions over the past decades. For example, the result is strengthened by considering the circular chromatic number, which is a real-valued lower bound on the chromatic number. The circular chromatic number of a quadrangulation cannot be in the interval (2,4). This parameter allows a generalization to larger even faces, for which a similar gap exists. In this work, we place these results into a framework based on the notion of winding number using extensions of colorings to continuous mappings. This yields unified and simplified proofs of gaps in the circular chromatic number for graphs with a distinguished set of directed even cycles. This generalizes the setting of graphs embedded on surfaces where every face is even. We further establish an analogous gap phenomenon when all faces are of a given odd length, previously known only in the case of triangulations. For example, we conclude that if G is a graph embedded on the projective plane such that all faces are 5-cycles, then either its circular chromatic number is 5/2 or at least 3, the former being the case only if G is Eulerian and every noncontractible facial walk is of odd length...

We define a notion of total positivity for the symmetric space $G/K$ by taking the Hausdorff closure of the image of Lusztig's totally positive part $G_{>0}$ in $G/K$. We introduce double Bruhat cells for the symmetric space and define their totally positive pieces. We prove a cell decomposition of the totally nonnegative symmetric space, give explicit positive parametrizations of all cells, establish closure relations, and show that the transition maps between the two natural families of parametrizations are subtraction-free.

The $k$-distance graph $G^k$ of a graph $G$ has the same vertex set as $G$ and two vertices are adjacent if and only if their distance is $k$ in $G$. These graphs have been extensively studied for their connection properties. In this paper, we study various properties of these graphs, including clique number and periodicity. For all $k$, we show that there exists a $k$-distance graph that is weakly periodic for any size period. Our main result is a proof that if $k > 2$ then there exists a $k$-distance graph of any size (strong) period. Finally, we provide evidence that there exists an upper bound on the periodicity of a connected $2$-distance graph as a function of $|V|$.

We establish a family of linear relations for the coefficients of the Tutte polynomial of a semimatroid. We also introduce ranked central sets, a mild generalization of semimatroids, and prove that the same identities hold in this broader setting. The proof relies on the specialization $x=\frac{z}{z-1}$, $y=z$ along the hyperbola $(x-1)(y-1)=1$; under this specialization, the edge subgraph expansion over central sets reduces to the $f$-polynomial of the central set complex. As applications, we obtain explicit formulas for Tutte polynomials of affine hyperplane arrangements and for balanced Tutte polynomials of biased graphs, whose central sets are central subarrangements and balanced edge sets, respectively. Our results recover the generalized Brylawski's identities for Tutte polynomials of ranked sets, matroids and graphs.

We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\log 4/\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\ge1$, the dimension-free mixed-alphabet sumset bound \[ |A+B|\ge (|A||B|)^{p_0}, \qquad A\subset\{0,1\}^d,\quad B\subset\{0,1,\ldots,m\}^d. \] For every $m\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger exponent is available.

A graph $G$ is $\mathcal F$-saturated if $G$ contains no member of $\mathcal F$, but the addition of any non-edge creates a copy of a member of $\mathcal F$. For $m\ge 1$, let $(m+1)K_3$ denote the vertex-disjoint union of $(m+1)$ triangles. In this paper, we study $(m+1)K_3$-saturated graphs. We construct a family of $(m+1)K_3$-saturated graphs which gives the uniform upper bound $\operatorname{sat}((m+1)K_3,n)=O(n^{3/2})$ for all $m\ge 1$ and $n\ge 3m+3$. For general $(m+1)K_p$-saturated graphs, Faudree et al. determined $\operatorname{sat}((m+1)K_p, n)$ for sufficiently large $n$. In the case of $p=3$, we lower Faudree's threshold from $n \ge 12m + 3$ to $n\ge 9m+5$ and prove that $\operatorname{sat}((m+1)K_3,n)=n+6m-1.$ We also provide two structural restrictions on the components obtained after deleting all vertices of degree $|V(G)|-1$ from an $(m+1)K_3$-saturated graph.

Solving large-scale instances of the Traveling Salesman Problem (TSP) exactly is computationally expensive. Researchers often employ graph sparsification methods to improve computational efficiency. Traditional sparsification methods typically rely on fixed heuristics and fail to fully exploit instance-specific structural information. In this paper, we propose Graph Edge Sparsification (GES), a learning-based sparsification approach for Euclidean TSP. By incorporating geometric structural information and combinatorial optimization technology, our proposed method adaptively generates a sparsification graph for different instances, significantly reducing the graph size and accelerating the solving process. Experimental results demonstrate that our sparsification method can prune up to 95% of edges on the MATILDA dataset, while keeping the solution gap within 1% of the optimal value. Moreover, our approach exhibits strong generalization capability on the TSPLIB benchmark.In some large-scale instances, the pruning rate exceeds 99%, while the optimality gap remains below 1%.

The binding number, denoted by $\mbox{bind}(G)$, of a graph $G$ is defined as the minimum value of $\frac{|N_G(X)|}{|X|}$ taken over any non-empty subset $X$ of $V(G)$ with $N_G(X)\neq V(G)$. A graph $G$ is said to be $r$-binding if $\mbox{bind}(G)\geq r$. The adjacency matrix of a graph $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$. An $H_b$-factor of a graph $G$ is defined as a spanning subgraph $F$ of $G$ such that for any $v\in V(G)$, $d_F(v)$ belongs to the set $\{1,3,5,\ldots,b-1,b\}$, where $b$ is an even integer with $b\geq2$. This note establishes a sufficient condition to guarantee that a connected $\frac{1}{b-1}$-binding graph $G$ of even order contains an $H_b$-factor based on the spectral radius.

Transfer systems as studied in equivariant algebra admit minimal generating sets, analogous to bases in linear algebra. It is natural to wonder if minimal generating sets form a matroid. We show that this happens only for lattices which are linear orders, or for cyclic groups of prime power order. In this case, we compute some invariants of the resulting matroid.

A star coloring of a graph $G$ is a proper coloring in which no path on four vertices is bicolored. The star chromatic number $\chi_{\star}(G)$ is the minimum number of colors in a star coloring of $G$. In this work we study star colorings from the perspective of forbidden induced subgraphs, focusing on three subclasses of chordal graphs. We provide both a structural characterization and a characterization in terms of forbidden induced subgraphs for star $3$-colorable chordal graphs; such characterizations yield a simple certifying recognition algorithm, running in time $O(|V|+|E|)$, for this class. We also characterize split graphs that are star $4$-colorable and star $5$-colorable in terms of (finitely many) forbidden induced subgraphs, again deriving linear-time certifying recognition algorithms. Finally, we study star colorings of $2$-trees and $2$-paths: we characterize the $2$-paths that are star $4$-colorable, prove that every $2$-path is star $5$-colorable, and exhibit a $2$-tree on $21$ vertices with star chromatic number $6$ such that any proper induced subgrahp has star chromatic number $5$.

The binary binomial sequences correspond to the diagonals of the Pascal's triangle modulo 2. They have interesting properties such as they form a basis of the linear space of all binary sequences with period a power of 2. Other properties of these sequences (period, linear complexity, construction rules or relations among different binomial sequences) have been deeply analysed in detail previously. In this work, we study the binomial $p$-ary sequences for a prime $p$, its intrinsic characteristic and formation rules. We also prove that the family of $p$-ary sequences with period a power of $p$ form a vector space over $\mathbb{F}_p$ and that the family of binomial $p$-ary sequences is a basis of this space.

For two vectors $x,y\in [b]^k$, consider the bipartite graph with two copies of $[b]$ in which $i$ on the left is joined to $j$ on the right if $(x_t,y_t)=(i,j)$ for some coordinate $t$. We study the largest size of a family $C\subseteq [b]^k$ such that, for every two distinct $x,y\in C$, this bipartite graph contains a cycle. We give a natural construction for such families and conjecture that it is optimal whenever $k$ is large relative to $b$. We prove an LYM-type upper bound that is asymptotically tight with respect to this construction, and is exact when $k$ is large and divisible by $b$. We then refine the argument using a circular ordering, obtaining the sharp full-support bound when $k\equiv -1\pmod b$. In the case $b=3$, we prove the exact general result when $k\equiv -1\pmod 3$ and $k$ is sufficiently large. The problem is motivated by the Daniely–Shalev-Shwartz dimension and the pseudocube formulation of a higher-alphabet Sauer-Shelah-Perles lemma.

We resolve Erdős Problems #593 and #1177. Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system; the answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Equivalently, after isolated vertices are removed, a finite triple system is obligatory precisely when it is linear, every hyperedge-node of its Levi graph has an incident bridge, and every Berge cycle is even. The proof uses an exact bridge-trace theorem for complete-rank one-apex sequence lifts. We also prove that, for every uncountable cardinal kappa, there is a linear triple system of chromatic number exactly kappa, with at most 2^{2^mu} vertices when kappa=mu^+. These two ingredients give a class-valued exact avoidance-spectrum dichotomy for every finite forbidden triple system. As a consequence, Erdős Problem #1177 has truth values yes, no, and yes. All results of this paper have been formally verified in Lean-4.

Let $g(r)$ be the minimum number of edges in an $r$-uniform intersecting hypergraph with cover number $r$. Erdős and Lovász proved the lower bound $g(r)\ge 8r/3-3$. We first give a completely elementary proof that $g(r)\ge 3r-4$. We then build on the same approach and apply Kahn's small-codegree hypergraph edge-colouring theorem to improve this to $g(r)\ge ((41-\sqrt{19})/12-o(1))r\approx 3.053r$. In particular, this shows that $g(r)>3r$ for all sufficiently large $r$, addressing a question of Erdős.