Papers

We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected $r$-uniform linear hypergraph with $r\ge3$ and nonnegative Lin–Lu–Yau curvature has this property. Among the various extensions of Ollivier–Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval $[1/2,1]$. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.

We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced in (Fink and Toghani, 2022). Our initial exploration examines the second- and third-order cases.

Let $M_n=M(K_{n+1})$ be the graphic matroid of the complete graph, and let $\mathcal{F}_k(M_n)$ be its rank-$k$ flats. We study families $\mathcal{A}\subseteq\mathcal{F}_k(M_n)$ satisfying $\mathrm{rk}(A\wedge B)\ge t$ for all $A,B\in\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erdős and L.~A. Székely~\cite{ErdosSzekelyHigher}. We prove the corresponding Erdős–Ko–Rado theorem in the explicit linear range $n+1\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\ge2k$. For every fixed $t$, we further prove an Erdős–Ko–Rado theorem under an explicit condition of order $O_t(k^2)$ on the block number $n+1-k$, with equality only for a full $t$-star. We also determine the largest nontrivial intersecting families under an explicit $O(k^6)$ threshold and characterize the unique extremal family up to isomorphism.

The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erdős, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.

In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

In this paper, we study the effective resistance, the Kirchhoff index, and the number of spanning trees of the connected graph $K_n-F$, which is obtained from the complete graph by deleting a set $F$ of $p$ edges. Let $B$ be the incidence matrix of the deleted edges. We call the matrix $Q=B^TB$ the edge-defect matrix. This is a $p\times p$ matrix which records, with signs, the way in which the deleted edges share their end vertices. First, we derive a formula for the effective resistance between any two distinct vertices in terms of the resolvent of the edge-defect matrix. This reduces the usual computation using the $n\times n$ Laplacian matrix to a computation using a $p\times p$ matrix corresponding to the number of deleted edges. Moreover, by using the eigenvalues of the same matrix, we give unified formulas for the Kirchhoff index and the number of spanning trees. Next, we derive a stability identity which exactly describes the excess from the Xu, Das, and Zhang type lower bound. As a consequence, we show that, in the range where a matching deletion can be realized, the Kirchhoff index is minimized when the deleted edge set is a matching. Furthermore, by using majorization, we prove that, for $p\ge 2$ and $n\ge \max\{4,2p-1\}$, among all non-matching deleted edge sets, the minimum is attained only when the deletion graph is isomorphic to $P_3\cup(p-2)K_2$. Finally, we apply the obtained formulas to several deletion graphs, such as matchings, stars, cliques, paths, and cycles.

For a fixed integer $k\ge2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $kK_3$-free graphs. We prove \[ \lambda(G)\le (k-1)+\sqrt{m-k(k-1)}. \] Moreover, equality holds precisely when $(2k-1)\mid m$ and, up to isolated vertices, $G$ is the join of $K_{2k-1}$ with an independent set of $m/(2k-1)-(k-1)$ vertices. The case $k=2$ was previously known; our argument establishes every fixed $k\ge3$. The proof requires information beyond first-order spectral stability. We derive an exact nonnegative defect identity at a maximum Perron vertex, use it to bound the entire outer layer by a constant, and reduce the remaining graph to a bounded core with finitely many independent twin classes. A Perron-vector concentration identity and the Erdős–Gallai matching theorem then force the unique extremal core. A nearly extremal family lies only $\Theta(m^{-1/2})$ below the target, showing why an exact second-order analysis is necessary.

For a finite set $A \subset \mathbb{R}_{>0}$ and a finite graph $H$, let $\chi_H(\mathbb{R}^n;A)$ be the minimum number of colors required to color $\mathbb{R}^n$ while avoiding a monochromatic copy of $H$ whose edges have distances in $A$. Extending the graph-copy framework of Axenovich, Liu, and Sagdeev and a multiple distance theorem of Naslund, we prove for any positive integer $m$, \[\chi_H(\mathbb{R}^n;m):=\max_{\substack{A \subseteq \mathbb{R}_{>0} \\ |A|=m}} \chi_H(\mathbb{R}^n;A) \geq \left(\Gamma_{\chi}\sqrt{\frac{m+1}{\Xi(H)}}+o(1)\right)^n.\] Here, $\Gamma_{\chi}$ is a constant and $\Xi(H)$ is an explicit structural parameter that can be substantially smaller than $|V(H)|-1$, thereby recovering Naslund's similar bound for complete graphs and improving the general bound inherited from the corresponding clique for many graph families. Along the way, we construct a weighted strengthening of the semi-diagonal flattening rank theorem of Correia, Sudakov, and Tomon.

Let $S$ be a grid in $\mathbb{Z}^d$. We wish to order $S$ so that the way two points compare depends only on the offset between them. Does each such order arise from a linear function? Bhattacharya conjectured, while Ren and Huang confirmed, that in two dimensions the answer is yes. On the other hand, wild orders do exist when $d \ge 4$. We settle the remaining case, by showing that the answer is yes in three dimensions as well.

Let $\A(2n)$ denote the set of up–down alternating permutations of $\{1,2,\ldots,2n\}$, and let \[ E_{2n}(q)=\sum_{\sigma\in\A(2n)}q^{\operatorname{inv}(\sigma)}. \] Andrews and Foata proved that $E_{2n}(q)\equiv q^{2n(n-1)}\pmod{(1+q)^2}$, and Liu recently obtained the cubic refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 \pmod{(1+q)^3}. \] Using the reciprocal generating function for the $q$-secant numbers, a third-order expansion of Gaussian coefficients at $q=-1$, finite differences, and Newton interpolation, we prove the fourth-order refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 +\binom n2(2n^2-2n-3)(1+q)^3 \pmod{(1+q)^4}. \] More generally, the recurrence yields an effective procedure for computing the expansion modulo $(1+q)^K$ for any prescribed $K$. We then apply the same local-expansion strategy to the generalized $q$-Euler numbers $E_{pn\mid p}(q)$ of Sagan and Zhang. For every prime $p$, we prove uniform congruences modulo $[p]_q^3$ and $[p]_q^4$; the fourth-order term is governed by a central $q$-Wolstenholme-type quotient associated with ${2p\brack p}_q$. Thus the fourth-order secant congruence is the first case of a general higher-cyclotomic method.

The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum over all vertices of a given function evaluated at the subgraph degree of that vertex. Here we study this problem and its complexity for quadratic functions. In particular, we introduce the critical roots of a graph, and show they define intervals over which the optimal value of the problem, as the quadratic root varies, is convex piecewise affine.

Let $\{0,1\}^{n}$ denote the set of all $n$-dimensional vectors whose components are either $0$ or $1$. For any two nonempty subsets $\mathcal{X},\mathcal{Y}\subseteq\{0,1\}^{n}$, the pair $(\mathcal X,\mathcal Y)$ is called an $n$-dimensional unique-sum pair if each pair $(\boldsymbol x,\boldsymbol y)\in\mathcal X\times\mathcal Y$ can be uniquely determined from the arithmetic sum $\boldsymbol x+\boldsymbol y$. Let $M(n)$ denote the maximum value of $|\mathcal X||\mathcal Y|$ over all $n$-dimensional unique-sum pairs $(\mathcal X,\mathcal Y)$. In 1969, Lindström proved that $$ \frac{1}{2}(1+\log_2 3) \leq \lim_{n\to\infty}\frac{1}{n}\log_2 M(n) \leq 3/2. $$ Since then, the lower bound $(1+\log_2 3)/2$ has been successively improved, whereas the upper bound $3/2$ has remained unchanged. In this paper, we establish an explicit upper bound whose numerical value is approximately $1.4884$. To the best of our knowledge, this is the first strict improvement over Lindström's upper bound $3/2$. Towards this end, we develop a coordinate projection approach that constructs a lower-dimensional unique-sum system from a unique-sum pair. By combining the results obtained from this approach with a relaxed form of a necessary condition established by Ordentlich and Shayevitz on unique-sum systems, we establish the above improved upper bound.

In this paper we derive new Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-locally recoverable codes (LRCs) with three attributes: first, the variable set is comparable in size to that of the classical Delsarte LP; second, our LP exploits the higher-order information forced by the local-distance condition through order \(\delta-2\), in the sense that for nondegenerate linear codes, its balanced base part gives exactly the same dimension bound as the symmetrized refined-weight LP of Gruica, Jany, and Ravagnani, while the additional constraints, nonvacuous whenever $\delta \ge 3$, give a further strengthening; and third, it applies to general $(r,\delta)$-LRCs, linear and nonlinear alike. Extensive computations over binary and ternary alphabets show that the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shortening and generalized Singleton bounds.

Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $\omega(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell}(n)=n$. When $\omega(\ell)\geq 2$, Bukh, Chao, and Zheng recently proved $\omega(\ell)n-O_{\ell}\left(n^{\frac{\omega(\ell)-2}{\omega(\ell)-1}}(\log n)^{C_{\ell}}\right)\leq f_{\ell}(n)\leq\omega(\ell)n-2\omega(\ell)\log n+11$ for some $C_{\ell}>0$. When $\ell$ has at least two distinct odd prime divisors, they further used Fourier analysis to improve the upper bound to $f_{\ell}(n)\leq\omega(\ell)n-(2\omega(\ell)+\varepsilon_{\ell})\log n$ for some $\varepsilon_{\ell}>0$, provided that $n$ is sufficiently large in terms of $\ell$ . For every fixed $\ell$ with $\omega(\ell)\geq2$, we prove \[ f_{\ell}(n)\leq\omega(\ell)n-\Omega_{\ell}(\log n\log\log n) \] for large $n$. The upper bound relies on a submatrix lemma of Bhowmick, Dvir, and Lovett, which is based on the bounded-torsion polynomial Freiman–Ruzsa conjecture recently proved by Gowers, Green, Manners, and Tao.

Let $\mathcal{P}$ be a three-dimensional convex polytope resting with one of its faces on the plane. At each step, $\mathcal{P}$ is allowed to roll over a randomly selected edge of the face currently lying on the plane, until the adjacent face comes to rest on the plane. The trace of $\mathcal{P}$ is the set of all points of the plane that can be reached by a vertex of $\mathcal{P}$, starting from a fixed initial position and performing a finite sequence of rolls. We prove that if the trace of $\mathcal{P}$ has a convergent subsequence, then, with probability one, the set of points reached by the vertices of a randomly rolling copy of $\mathcal{P}$ is everywhere dense in the plane. This settles a conjecture of Hegyvári.

We characterize four groups of Clifford circuits for Calderbank–Shor–Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.

We prove that the Numerical Terao Conjecture holds for even-degree conic-line arrangements having only ADE singularities. We then show that the conjecture fails in the broader quasi-homogeneous setting once ordinary quadruple points are admitted, by constructing a degree-nine counterexample consisting of a pair of conic-line arrangements, each with seven lines and one smooth conic, with the same weak combinatorics \[ W(\mathcal{CL}) = (7,1;\,8A_1+D_4+4X_9). \] We show that one curve is free with exponents $(4,4)$, whereas the other is nearly free with exponents $(3,6)$. This yields a counterexample to the strongest known formulation of the Numerical Terao Conjecture.

The genus polynomial of a graph is the generating polynomial for the number of nonequivalent embeddings of the graph on each orientable surface. In this paper, we address three questions on genus polynomials for wheel graphs: the computation of genus polynomials, the unimodality and the asymptotic normality of their coefficients. We derive an explicit formula for the genus polynomial of wheel graphs by combining methods of the joint tree model and characters theory, and then prove its real-rootedness. This stronger result implies the log-concavity, unimodality, and asymptotic normality of its coefficients. Thus, we confirm the unimodality conjecture for the genus distribution of wheel graphs and provide a positive answer to the asymptotic normality question posed by Zhang, Peng, and Chen (Adv. in Appl. Math. 127 (2021), 102175).

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $\beta(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $\alpha(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\geq4$, we prove $\beta(d,n)=n-\Theta(n^{1/\lfloor d/2\rfloor})$. For flag spheres, we show $\alpha(d,n)\geq n-4\sqrt n+O(1)$ for all $d\geq4$ and determine the correct asymptotic order $\alpha(d,n)=n-\Theta(\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to our other results, for this very large family of spheres, the independence number cannot be larger than $\left\lfloor\frac{n}{2}\right\rfloor.$

Nguyen~\cite{Nguyen2025} recently proved that every $\{P_5,C_5\}$-free graph $G$ satisfies $\chi(G)\leq \omega(G)^{40}$. Building on his framework, we introduce two refinements, namely a sharper cutset decomposition using the $C_5$-free condition and an improved density-increment argument. These yield a polynomial $\chi$-binding function with exponent $24$, improving the previous bound of $40$.

A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into queues so that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue number. The planar product structure theorem states that every planar graph is a subgraph of the strong product of a graph of simple treewidth at most $3$, a clique $K_3$, and a path. Such a strong product admits a queue layout with $49$ queues (Wood, 2005), which implies that the queue number of planar graphs is at most $49$. Recently, Bekos, Gronemann, and Raftopoulou (Algorithmica, 2023) investigated how the general approach based on the product structure can be optimized for planar graphs. They claim that by appropriately reordering the three vertices in each bag arising from a tripod, it is possible to reduce the queue number of planar graphs to~$42$. In this note we highlight a gap in their queue layout algorithm: one of the choices required by the algorithm is not guaranteed to exist. Hence the claimed upper bound of $42$ queues is not established by the published proof.

We establish several new series evaluations involving the central $q$-binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the limiting case at $q\to 1$.

For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers–Ramanujan and Andrews–Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers–Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.

For a graph G and a nonempty set S of vertices, the edge multiset representation of an edge e is the multiset of distances from e to the elements of S, where d(uv,s)=min{d(u,s),d(v,s)}. The edge multiset dimension edim_m(G) is the minimum cardinality of a set whose edge representations are pairwise distinct, and is infinite if no such set exists. A recent survey asked whether edim_m(Q_d) is infinite for every d >= 3. We answer this question negatively and determine the finite-infinite transition completely: edim_m(Q_d) is infinite if and only if 2 <= d <= 5. An exhaustive computation proves edim_m(Q_5) = infinity, extending the known nonexistence results for Q_3 and Q_4. Explicit independently verifiable resolving sets are given for Q_6 through Q_10. For all d >= 11 we prove existence probabilistically: equality of two random edge histograms is a zero-divergence event on a graph of distance levels, and conditioning outside a spanning forest bounds its probability by a product of central-binomial atoms. Certified exact rational computations cover 11 <= d <= 50, and an elementary ten-edge forest estimate handles the tail d >= 51. We also prove the lower bound edim_m(Q_6) >= 6.

We study the following problem: given an edge-colouring of a $K_n$ with $r$ colours, how many colours does one need to keep (in the worst-case scenario) so that the subgraph formed by those colours is connected? We determine this extremal function up to constant factors in every parameter regime, and show that it has the same asymptotic order as the corresponding number (of colours) needed for ensuring that no vertex is isolated. This gives a complete description of the “spanning” threshold for connectivity in edge-coloured complete graphs.

In a graph, a set $D$ is $k$-dominating if every vertex in $V(G) \setminus D$ has at least $k$ neighbors in $D$. Jafari, Alikhani, and Bakhshesh introduced the concept of a $k$-coalition, which is a pair of disjoint sets $X_1$ and $X_2$ of vertices such that neither is a $k$-dominating set but $X_1 \cup X_2$ is a $k$-dominating set. A $k$-coalition partition is a vertex partition in which each set either forms a $k$-coalition with some other set or is itself a $k$-dominating set with exactly $k$ vertices. The $k$-coalition number $\operatorname{CO_k}(G)$ is the maximum number of sets in a $k$-coalition partition. We compute the $k$-coalition number for several families and bound the $k$-coalition number under disjoint union and graph join. We show that the set of possible sizes of $k$-coalition partitions forms an interval. Finally, we investigate $k$-coalition graphs and prove that every graph is a $k$-coalition graph.

A latin square of order $n$ can be viewed as a partition of the $n \times n$ all-ones matrix into permutation matrix summands. Here, we consider a relaxation in which the matrix summands are allowed to be induced from shorter permutations. For $\sigma \in S_k$, the `fuzzy permutation matrix' $P_\sigma^{\uparrow n}$ arises from combining all $\binom{n}{k}^2$ order-preserving embeddings of the $k \times k$ permutation matrix $P_\sigma$ into an $n \times n$ matrix. We define a fuzzy latin square as a linear combination of $n \times n$ fuzzy permutation matrices $P_\sigma^{\uparrow n}$ equaling a constant matrix. We study various aspects of these objects, including certain relevant vector space dimensions and a census of fuzzy latin squares with a small number of terms. In particular, we determine strong conditions on four-term fuzzy latin squares in the `vanishing' case (when the constant matrix is all zeros). We also report on a computer-assisted classification of six-term fuzzy latin squares in the non-vanishing case.

Edge open packing is a relaxation of induced matching in which the selected edges may induce disjoint stars. We study the Maximum Edge Open Packing problem on permutation graphs, interval graphs, and well-partitioned chordal graphs. For the first two classes, we introduce an oriented star-conflict graph whose vertices are ordered edges. We prove that its compatibility graph admits a natural transitive orientation: a product-order orientation for permutation graphs and a left-to-right orientation for interval graphs. In each case, a maximum edge open packing is obtained from a maximum clique, equivalently a longest directed path, in the compatibility graph. Given the corresponding representation, both algorithms run in \(O(n^2+m^2)\leq O(n^4)\) time, where \(n=|V(G)|\) and \(m=|E(G)|\). For well-partitioned chordal graphs, we give a dynamic program over a partition tree. Its states use the fact that the endpoint set of an edge open packing meets each clique bag in at most two vertices. Given a partition-tree representation, the edge open packing number is computed in \(O(n^4)\) time, and an optimal packing can be reconstructed within the same time bound.

For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible Néron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.

In 2016, Berkovich and Uncu proved that, for all nonnegative integers $i$, $j$, and $n$, the number of strict partitions of $n$ with $i$ odd-indexed odd parts and $j$ even-indexed odd parts equals the number of strict partitions of $n$ with $i$ parts congruent to $1$ modulo $4$ and $j$ parts congruent to $3$ modulo $4$. Their proof used generating functions, and they asked for a combinatorial proof. We answer their question with an explicit bijection, assembled from three classical ingredients: $2$-modular diagrams, an insertion algorithm of Chen, Gao, Ji, and Li, and Glaisher's bijection. AxiomProver autonomously formalized and verified the proof of the main theorem in Lean.