Papers

For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

We show that Makhlin's polytopes in type $B_n$ are unimodularly equivalent to string polytopes for Demazure modules in type $B_{2n-1}$. The proof passes through type $C$, combining Makhlin's diagonal comparison with symplectic FFLV polytopes and the $B/C$ similarity for string cones.

We construct an Orlik–Solomon algebra for any locally geometric poset as a natural generalization of the one for geometric lattices. This algebra has several interesting features, including a combinatorial no-broken-circuit basis that we obtain through Gröbner basis theory. When the poset captures the intersection data of an arrangement of certain subgroups in a complex abelian Lie group, we infuse the Orlik–Solomon algebra with topological information to compute the cohomology of the arrangement complement. In particular, when the Lie group is compact, we present an explicit differential graded algebra whose cohomology is the rational cohomology of the arrangement complement. We further exhibit a family of abelian arrangements whose integer cohomology has torsion.

Suppose we are given graphs $B$ and $G$. In the classical Maker-Breaker game $\text{MB}(B,G)$ two players, Maker and Breaker, alternately claim edges of $B$ and it is Maker's goal to claim a copy of $G$ in $B$, while it is Breaker's goal to prevent that. In this paper, $B$ is the countably infinite complete graph $K_{\aleph_0}$ and we are given finitely many infinite subgraphs $G_1, \dots, G_k \subseteq B$. In the color preserving game, it will be Maker's goal to claim a $K_{\aleph_0} \subseteq B$, which contains infinitely many edges of each $G_i$. We present sufficient winning conditions for both Maker and Breaker, if $k > 1$ and a full characterization of the game, if $k =1$. This partly answers a question of Bowler, Emde and Gut. In the (partially) pattern preserving game, it is Maker's goal to claim a copy $K$ of $K_{\aleph_0}$, such that $G_i \cap K$ is isomorphic to (a subgraph of) $G_i$ for all $i \in [k]$. In those games, we investigate some patterns for which Maker has a winning strategy.

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.

We construct a smooth strictly positive density in $\mathbb{R}^4$ that cannot be divided into $16$ parts of the same size by four affine hyperplanes. This settles the last open case of Grünbaum's 1960 conjecture and disproves Ramos' general conjecture on hyperplane equipartitions. We reduce the construction to finding two homogeneous polynomials in four variables, of degrees three and four, whose multilinear coefficients cannot vanish simultaneously after any orthogonal change of coordinates. We give two proofs of this nonvanishing result. The first uses a local perturbation argument. The second reduces it to the absence of a common zero for five explicit polynomials on $[-1,1]^6$, verified by a computer-assisted Bernstein subdivision argument.

We consider a Turán-type problem arising in phylogenetics: determining the maximum size of a 2-weakly compatible split system. This compatibility condition arises in the reconstruction of phylogenetic networks from quartet weights. It was previously shown that a 2-weakly compatible split system has size at most \[ 3\binom{n}{4}+\binom{n}{2}. \] We prove that the maximum size is $O(n^{5/2})$.

In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Brändén and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In this note, we make the next step by proving it for $M_2^\natural$-concave functions. This shows the same property for the intersection of any pair of matroids (which itself may not be a matroid). Furthermore, we show that this can not be further extended to the intersections of three matroids by including a counterexample of partition matroids.

In 1979, Albertson and Berman conjectured that every planar graph $G$ contains an induced forest of order at least $|V(G)|/2$. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematical arguments and exhaustive computations to show that the minimum order of a counterexample is $29$. We also construct infinitely many $4$-connected $5$-edge-connected counterexamples (and show that the unique such counterexample of minimum order has order $41$), whereas previously all known counterexamples had vertex-connectivity at most $3$. Furthermore, we construct an infinite family of planar graphs on $n$ vertices whose maximum induced forests have order at most $\frac{25}{52}n$, thereby improving the previous best upper bound. This family also yields infinitely many counterexamples (for every integer $d \geq 7$) to a conjecture of Chappell and Pelsmajer concerning induced forests of maximum degree at most $d$.

Let $\beta$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $\beta$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $\beta$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.

Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin–Lu–Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin–Lu–Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin–Lu–Yau curvature does not force higher-dimensional GLMY path homology to vanish.

An explicit sequence of tetrahedra generated by the longest-edge bisection algorithm is shown to degenerate. The example violates shape regularity and both the minimum- and maximum-angle conditions, demonstrating that arbitrary tie-breaking among longest edges does not guarantee nondegeneration.

Tuza's conjecture states that for any graph $G$, the minimum size of a triangle transversal $\tau(G)$ is at most twice the maximum size of a set of edge-disjoint triangles $\nu(G)$. In this note, we prove $\tau(G) \leq \frac{63}{22}\nu(G)$, improving the previous bound $\tau(G) \leq \frac{66}{23}\nu(G)$ established by Haxell in 1999. The key observation is that for a "2-colorable" family of triangles $\mathcal{F}$, where each triangle has two blue edges and one red edge, we can obtain $\tau(\mathcal{F})\leq (1+\sqrt{3})\nu(\mathcal{F})$.

Let $q$ be an odd prime power, and $\mathcal{C}, \mathcal{D}$ be two smooth plane conics defined over $\mathbb{F}_q$ with transversal intersection. We present an exact formula for the number of points in $\mathbb{P}^2(\mathbb{F}_q)$ that are internal/external to $\mathcal{C}$ and internal/external to $\mathcal{D}$. This refines the $\frac{q^2}{4}+O(q^{3/2})$ asymptotic estimate for this quantity due to Asgarli and Yip \cite[Theorem 1.2]{Asgarli}. In particular, we show that the error term is of size at most $q+\sqrt{q} + 1$. By studying the geometry of the incidence variety related to this problem, we link the exact point counts directly to the Frobenius traces of two associated elliptic curves, and the number of $\mathbb{F}_q$-rational intersection points of $\mathcal{C}$ and $\mathcal{D}$ and of the corresponding dual conics $\mathcal{C}^*$ and $\mathcal{D}^*$. Lastly, We provide a remark explaining the challenges in generalizing this method to the study of higher-dimensional quadrics.

Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=\pi\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy–Ramanujan formula for integer partitions.

Signed difference sets (SDSs) extend ordinary difference sets by allowing negative coefficients. We establish new nonexistence criteria and existence constructions for SDSs in finite abelian groups. Using the classical self-conjugate-prime method and a support-refined quotient method, we derive four obstructions, including the $C_2$-quotient, near-full-support, and small-defect obstructions. Applied cumulatively to the $67{,}823$ open group-specific cases in the database with $9\leq v\leq499$, these criteria rule out $45{,}361$ cases. For existence, we construct an infinite family from PCP-type regular partial difference sets arising from Desarguesian spreads and an explicit $(125,28,3)$-SDS in $C_5^3$ using quartic multiplicative characters. These constructions settle three further open cases.

The List Edge-Coloring Conjecture predicts that any graph whose edges can be colored with $k$ colors can also be colored from arbitrary lists of $k$ colors. We prove its stronger online form for two new infinite families, $K_{p-1}$ and $K_{2p}$, where $p$ is an odd prime. For even $n$, order the vertices of $K_n$ and draw each perfect matching as arcs above them. Count crossings separately within each matching, and let $S_n$ be the number of decompositions into perfect matchings having an even total crossing count minus the number having an odd total. Then \[ S_{p-1}\equiv\left(\frac{-2}{p}\right)\pmod p, \qquad S_{2p}\equiv-p\pmod {p^2}. \] The two congruences are governed by the same elementary matching sum over $\F_p$, although their proofs use the prime $p$ differently. Their nonzero residues give the conjectured values even in the online game. They also treat the corresponding complete graphs with one perfect matching removed, as well as $K_{2p}$ after deleting some, but not all, of a natural cyclic family of $p$ disjoint perfect matchings.

For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l>k\ge 2$, $$ \text{ex}(n,C_{2k+1},\mathscr {C}_{2k}\cup\{C_{2l+1}\}) =O_{k,l} \left(n^{2-\frac{1}{k(k+1)(l-k)}}\ \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Győri, Methuku, and Vizer.

A graph $G$ is apex if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.

In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J: edges among vertices with the same parent follow G, while edges between vertices whose parents are horizontally linked follow J. The substitution graph is defined as its underlying graph. Substitution graphs provide a purely combinatorial model of self-similar structures, independent of any underlying geometric structure. Furthermore, we establish a necessary and sufficient condition for substitution graphs to be hyperbolic, formulated in terms of the vanishing of path matrices associated with sufficiently long shortest horizontal paths. Based on this characterization, we further derive several conditions that are either necessary or sufficient for hyperbolicity, depending only on the generators G and J.

Let $0\le \mu_1\le \mu_2 \le \cdots \le \mu_n$ and $\delta > 0$. Samuels' conjecture claims that if $X_1,\dots,X_n$ are independent non-negative random variables with $\mathbb{E}[X_i] = \mu_i$, then $$ \mathbb{P}\left( \sum_{i=1}^n X_i < \delta + \sum_{i=1}^n \mu_i \right) \ge \min_{1\le i\le n} \prod_{j=i}^n \left(1-\frac{\mu_j}{\delta + \sum_{k=i}^n \mu_k}\right).$$ This conjecture was recently proved by Ling. In this note, we provide an alternative presentation of Ling's proof via an operation called the cap move. This proof works directly with finitely supported distributions and avoids the reduction to the Bernoulli case.

A triangulation is a graph on a surface where every face is triangular. It is well-known that a planar triangulation $G$ is $3$-chromatic if and only if $G$ is even, that is, all the vertices of $G$ have even degree. We focus on even triangulations of non-spherical surfaces. It is known that there is an invariant of even triangulations, called a monodromy. This concept was firstly introduced by Hutchinson et al. [J. Combin. Theory Ser. B 84 (2002) 225–239], and the congruence class of monodromies was defined by Kawarabayashi et al. [J. Combin. Theory Ser. B 99 (2009) 229–246]. For a lower genus surface $F^2$, the number of congruence classes of monodromies have already been shown, $1$ if $F^2$ is the sphere, $2$ if $F^2$ is the projective plane by Mohar [Discrete Math. 244 (2002) 339–343] and $3$ if $F^2$ is the torus by Higuchi et al. [Discrete Math. 311 (2011) 1128–1135]. In this paper, we count the number of congruence classes of monodromies of even triangulations of general closed surface $F^2$.

Based on work with Greenfeld and with Ziegler, Tao showed a concatenation result that if a map $f : \mathbb{F}^2 \to \mathbb{F}$ is affine-linear on every line parallel to the coordinate axes, and on all lines with a fixed nonzero slope (where the field $\mathbb{F}$ has size $> 2$), then $f$ is affine-linear on $\mathbb{F}^2$. We extend this from $\mathbb{F}^2$ to $\mathbb{F}^n$ and obtain a tight minimum number of additional lines needed – $N = \binom{n}{\lfloor n/2 \rfloor}$ – for every field $\mathbb{F}$ with $3 \leqslant n < |\mathbb{F}|$. The proof is constructive and shows a stronger result: the existence of a universal family of $0$-$1$ matrices of size $\binom{n}{k} \times \binom{n}{k}$ (one for each pair $0 \leqslant k \leqslant n$), which are indexed by ballot sets and are unimodular over all unital commutative rings. We also show a second tightness: of the assumption $n < |\mathbb{F}|$. Else, there exist multi-affine maps $f$ which are affine-linear on every line through the origin, but not affine-linear globally on $\mathbb{F}^n$. More strongly, we prove this dichotomy – including the bound of $N$ – over all integral domains, or Noetherian (e.g.\ finite or Artinian) rings, or products of these. This yields a novel numerical invariant for affine-linearity, for every product of Noetherian rings and integral domains.

Lovász and Szegedy asked whether a nonconstant continuous, or even smooth, finitely forcible kernel exists. For every $0<\lambda\leq1/128$, we construct an explicit nonconstant real-analytic kernel $W_\lambda$ taking values in $(1/4,7/8)$. A single finite family of simple graphs, independent of $\lambda$, forces every $W_\lambda$ among all bounded symmetric real-valued kernels on $[0,1]^2$.

Given graphs $H$ and $F$, the generalized Turán number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Turán problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $\chi(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=\Omega(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $\chi(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).

We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm $\gamma$ admits a signed decomposition into at most $2^{O(\gamma^4)}$ elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from $\gamma^6$ to $\gamma^4$.

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/\epsilon)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic graph has the same numbers of balanced and unbalanced vertices.

An equivalent formulation of the Riemann hypothesis recently led to a partition expansion naturally indexed by diagonals of the Young lattice. Segovia isolated the hook families $(r,1^m)$ on these diagonals and computed their limiting contributions $\rho_r$, while observing that non-hook families provide a missing contribution. We introduce a bivariate finite generating function that packages all Young shapes on every fixed-excess diagonal at once. For each fixed $r\geq1$, we obtain a diagonal generating polynomial $D_r(n;z)$ and prove \[ A_r(n)\sim C_r\,n\log\log n, \] where $C_r$ is the $(r-1)$st coefficient of an explicit convergent infinite product. Moreover, \[ C_r=\sum_{\nu\vdash r-1} C_\nu , \] giving a canonical decomposition over the partitions of the excess $r-1$. The one-part contribution is Segovia's hook constant $\rho_r$, while the remaining terms give all non-hook corrections simultaneously. We then refine these constants by introducing coefficients $C_{r,d}$ that record simultaneously the Young-lattice excess $r-1$ and the number $d$ of non-unit rows. Row sums recover the fixed-excess constants $C_r$, while column sums recover the depth decomposition in an Abel-regularized multiple-zeta expansion of $e^\gamma$. More precisely, each partition $\nu\vdash r-1$ is identified with an Abel-regularized multiple-zeta block of depth $\ell(\nu)$. Thus the same array $(C_{r,d})$ organizes the decomposition simultaneously by Young-lattice excess and multiple-zeta depth. Our results concern the combinatorial and asymptotic structure of this decomposition, rather than the Riemann hypothesis itself.

We consider families of lines that cover every point of the integer lattice $\mathbf{Z}^2$ in the plane, subject to the constraint that no two lines of different direction in the family meet at a lattice point. Restricting to lattice lines (lines containing at least two, hence infinitely many, lattice points, equivalently of rational direction), we show that the set of directions occurring in such a covering can be made dense in the space of line directions. The construction is a recursive splitting of $\mathbf{Z}^2$ into nested rank-2 sublattice cosets, each handed off to a freshly chosen direction; the key technical point is a steering lemma showing that at every stage of the recursion a new direction arbitrarily close to any prescribed target can still be realized, via an elementary sieve bound.