Papers

We show that matroid Stanley-Reisner rings can have essential deformation obstructions of arbitrarily high order over every field, even at corank three. For every $m\ge3$, we construct a connected cosimple matroid with $m$ fine-homogeneous first-order deformation classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. These classes define a joint flat family whose $(m-1)$-jet has no lift to order $m$, even without preserving the grading. For the duals of rank-three truncations of $U_{2,m}\oplus U_{2,n}$, $m,n\ge3$, every nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and this quotient has an explicit module presentation with $m+n-2$ minimal homogeneous generators. The construction is based on a computation of the full multigraded second cotangent cohomology $T^2$ and its primary cup image in terms of flats and parallel classes of the dual matroid. This computation also proves that $T^2$ vanishes precisely when every connected component has corank at most two, equivalently when the matroid has no $U_{1,4}$-minor, settling the conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello. The same criterion characterizes formal smoothness of the affine deformation functor.

Given graphs $X,Y$ on $n$ vertices, Defant and Kravitz defined the friends and strangers graph, denoted $\mathrm{FS}(X,Y)$. The vertices of $\mathrm{FS}(X,Y)$ are bijections $\sigma:V(X)\to V(Y)$, and two bijections $\sigma$ and $\sigma'$ are adjacent in $\mathrm{FS}(X,Y)$ if they differ by a transposition $(ab)$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$. Our main result resolves a conjecture of Bangachev that if $X,Y$ are connected graphs and $2\min\{\delta(X),\delta(Y)\}+3\max\{\delta(X),\delta(Y)\}\ge 3n$, then $\mathrm{FS}(X,Y)$ is connected. The main step in the proof is showing that there exists an absolute constant $C$ such that for any adjacent vertices $a,b\in V(X)$ and any vertex $\sigma\in V(\mathrm{FS}(X,Y))$, $a$ and $b$ can swap positions in at most $C$ moves. Furthermore, we prove that if $X,Y$ are connected graphs satisfying the above minimum degree condition, then the diameter of $\mathrm{FS}(X,Y)$ is $O(n^2/\min\{\delta(X),\delta(Y)\})$, and that this is sharp as a bound on the maximum diameter of $\mathrm{FS}(X,Y)$ in this regime. This work improves the diameter results of Akella and Li, and resolves many of their questions. Moreover, we show that even for disconnected graphs $X$ and $Y$, the same minimum degree condition implies that the diameter of any connected component of $\mathrm{FS}(X,Y)$ is $O(n^2)$.

We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

The independence complex of the Kneser graph $\KG(n,k)$ is the simplicial complex of pairwise intersecting families of $n$-subsets of $[2n+k]$; equivalently, it is a Vietoris–Rips complex of the $n$-subsets of $[2n+k]$ under the symmetric-difference metric. Feng and Wang asked whether, for $n=3$, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every $k\ge0$, the only nonzero reduced Betti numbers using rationals of $\Ind(\KG(3,k))$ are \[ \beta_6=\binom{k+5}{7}+29\binom{k+6}{7} \qquad\text{and}\qquad \beta_9=\binom{k+6}{6}. \] The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer–Vietoris spectral sequence which reduces the problem to the six-element sets.

Oda's smoothness conjecture asserts that every smooth lattice polytope has the integer decomposition property. The three-dimensional case of this conjecture was recently proven by AI. Here we give a much shorter proof of this result.

Every 3-regular graph of girth at least $g$ has a cut containing at least a fraction $M_g$ of its edges, and the best rigorous bounds at large girth are $0.912\le \lim_{g\to\infty}M_g \le 0.92386$. Lower bounds on $M_g$ follow from the performance of local algorithms: on a graph of girth $g \ge 2p+2$ a depth-$p$ local algorithm sees the same tree around every edge, so its expected cut fraction on that tree bounds $M_g$ from below. We apply this strategy to the Variational Iterative Rotation Algorithm (VIRAL), a classical local algorithm, and evaluate it on the depth-$p$ tree for every even depth up to $p=32$. The resulting bounds exceed those recently obtained from the Quantum Approximate Optimization Algorithm (QAOA) for every $p\ge 4$, by $0.005$ to $0.011$, and they extend the range of girths covered from $g=36$ to $g=66$. Notably, at $g=66$ we obtain $M_{66} \ge 0.9127$ at $99.99999\%$ confidence level, which raises the asymptotic lower bound from $0.912$ to $0.9127$.

Fix integers $k,b,q$ with $k\ge 2,b\ge 0,q\ge 2$. Define the function $p$ to be: $p(x)=kx+b$. We call a set $S$ of integers $(k,b,q)$-linear-free if $x\in S$ implies $p^i (x)\notin S$ for all $i=1,2,\dots,q-1$, where $p^i (x)=p\left(p^{i-1}(x)\right)$ and $p^0 (x)=x$. Such a set $S$ is maximal in $[n]:=\{1,2,\dots,n\}$ if $S\cup \{t\},\forall t\in [n]\backslash S$ is not $(k,b,q)$-linear-free. Let $M_{k,b,q} (n)$ be the set of all maximal $(k,b,q)$-linear-free subsets of $[n]$, and define $g_{k,b,q} (n)=\min_{S\in M_{k,b,q} (n)}|S|$ and $f_{k,b,q} (n)=\max_{S\in M_{k,b,q} (n)}|S|$. In this paper, formulae for $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$ are proposed. Also, it is proven that there is at least one maximal $(k,b,q)$-linear-free subset of $[n]$ with exactly $x$ elements for any integer $x$ between $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$, inclusively.

A classical theorem of Kaiser and Rabinovich implies that if every subfamily of size at most $\lfloor\log_2(d+1)\rfloor+1$ of a finite family of compact connected sets in $\mathbb{R}^d$ has a common axis-parallel hyperplane transversal, then the entire family has an orthogonal star transversal, i.e., a union of $d$ hyperplanes, one perpendicular to each coordinate axis. We study colorful variants of this result. First, we use standard KKM machinery to establish a colorful analogue with $d+1$ color classes and a $\lfloor\log_2(d+1)\rfloor+1$ transversal hypothesis for rainbow subfamiles. Then we focus on reducing the required number of colors. To this end, we introduce the rainbow pushability $\pi_r(P)$ of a polytope and we use it to state and prove few-color variants of the colorful Carathéodory theorem and the colorful polytopal KKMS theorem. Applying the latter to the $d$-dimensional cube, we show that $\lfloor\log_2 d\rfloor+2$ color classes suffice for our transversal problem, provided that every rainbow choice has a common axis-parallel hyperplane transversal. We also extend both colorful results to the setting of convex $(n,D)$-bodies and we show that three colors are needed for $d\geq 2$.

The Hilbert–Poincaré series of the Chow rings of the toric varieties of matroids, often called the Chow polynomials, and the $f$-polynomial of their boundary complexes, called the chain polynomials, have been conjectured to only have real zeros. We construct several explicit infinite families of matroids from finite geometries, whose multichain counts fail to be log-concave. They share a common geometric construction called hyperplane coalescence, together with several classical operations in projective geometry and combinatorial designs. This induces failures of log-concavity for Chow, augmented Chow, chain, and $h$-polynomials of several explicit infinite families of matroids, disproving the real-rootedness conjectures of Ferroni–Schröter, Huh–Stevens, and Athanasiadis–Kalampogia-Evangelinou from the lowest possible ranks. As an application of the same mechanism, we also give counterexamples to the log-concavity of $h$-polynomials of graph associahedra, answering in the negative a question of Sack.

We characterise all (undirected) cacti graphs, and forests of cacti, with an edge-preserving near-unanimity operation, by means of forbidden induced subgraphs. Indeed all such graphs also have an edge-preserving majority operation. We show that said majority can be chosen to be symmetric and explicitly construct it for all forests of cacti without the forbidden induced subgraphs.

We study block sensitivity and certificate complexity of rotation symmetric Boolean functions, with particular emphasis on monomial rotation symmetric Boolean functions. We first show that every odd-degree monomial rotation symmetric Boolean function on $n$ variables satisfies $s=bs=C=n$. We then determine sensitivity, block sensitivity, and certificate complexity for every quadratic monomial rotation symmetric Boolean function. For the quadratic function $Q_{n,k}$ generated by $x_0x_k$, we prove, outside the short-orbit case, that \[ C(Q_{n,k})= \begin{cases} n,&4\mid n/\gcd(n,k),\\ n-\gcd(n,k),&4\nmid n/\gcd(n,k). \end{cases} \] Moreover, \(bs(Q_{n,k})=C(Q_{n,k})\) for every quadratic MRS function, and outside the short-orbit case, strict separation \(s<bs=C\) occurs exactly when \[ \frac{n}{\gcd(n,k)}\equiv2\pmod4. \] Finally, for the consecutive quartic family \[ R_n(x)=\bigoplus_{i\in\mathbb Z_n}x_ix_{i+1}x_{i+2}x_{i+3}, \] we determine all three measures in every residue class modulo $6$, using the sensitivity formulas and, in one residue class, finite-state results established in our companion preprint~\cite{LiZhangRSBFSensitivity}. In particular, for $r=2,3$ and $6q+r\ge5$ we obtain strict separation, \[ \boxed{s(R_{6q+r})=6q<bs(R_{6q+r})=C(R_{6q+r})=6q+1,\qquad r=2,3.} \] Thus this family gives an infinite collection of rotation symmetric Boolean functions for which sensitivity is strictly smaller than block sensitivity, while block sensitivity and certificate complexity coincide.

Let $X$ be a connected graph and let $\lambda_1(X)$ denote the spectral radius of its adjacency matrix. An edge $e$ of $X$ is sustainable if it can be deleted and replaced by a non-edge so that the resulting graph is connected and its spectral radius does not decrease; $X$ is sustainable if every edge is. We study this edgewise notion of spectral robustness for trees, where each replacement is an elementary exchange between trees on the same vertex set. We establish basic properties of sustainability and exhibit infinite families of sustainable and non-sustainable trees. Our main results concern trees with a vertex of large degree: if $T$ is a tree on $n$ vertices with $\Delta(T)\geq n/2$, then every vertex of maximum degree is a principal vertex, that is, a vertex at which the Perron vector is maximal. Building on this, we prove that every tree with $\Delta(T)\geq n/2+1$ is non-sustainable, and we classify the sustainable trees in the boundary case $\Delta(T)=n/2$: for $n\geq 4$, the only such trees are the double broom $B(2,n/2,n/2-1)$ and one exceptional tree on eight vertices. We conclude with open problems on the generalized tree-shift order and the asymptotic proportion of sustainable trees.

Let $q$ be a prime power. An $(r, i)$-regular fat $\mathbb F_q$-linear set has exactly $r$ points of weight $i\ge2$, called fat points. All of its remaining points have weight one, and at least one such point occurs. We solve two open problems posed by three of the authors of this paper. First, we derive divisibility constraints on the point weights using the scalar stabilizer of the defining subspace. Consequently, a regular fat linear set of rank $2i$ in $\operatorname {PG}(1,q^n)$ has at most two or exactly $q + 1$ fat points. In the latter case, the fat points form an $\mathbb F_q$-subline. Second, we construct higher-dimensional regular fat linear sets whose fat points do not form an $\mathbb F_q$-subgeometry. In particular, for every $k\geq 3$ we obtain examples of rank $2k$ whose fat locus consists of an $\mathbb F_q$-subgeometry $\operatorname {PG}(k-2,q)$ in $\operatorname {PG}(k-1,q^4)$ together with one external point. This construction is then generalized further. Finally, we apply our projective-line results to $q$-matroid representability and present an infinite family of non-representable rank-two $q$-matroids arising from partial spreads.

Rotation symmetric Boolean functions are Boolean functions invariant under cyclic permutation of their variables and can be represented by binary necklaces. We express local sensitivity as a weighted bichromatic degree in the necklace quotient and derive an exact formula for the multiplicities between neighboring necklaces. For an odd prime number of variables, we obtain explicit sensitivity formulas and show that multiplicity corrections occur only on an asymptotically negligible family of necklaces. For monomial rotation symmetric Boolean functions, we prove that every odd-degree function has maximum sensitivity, determine the sensitivity of every quadratic function, and determine the sensitivity of the consecutive quartic family. We also determine exactly the sensitivity of cyclic run-threshold functions for arbitrary length and threshold.

We introduce a determinantal linear process on the space of $1$-forms over a weighted graph endowed with a unitary local system, which we call the determinantal linear spanning forest (DLSF). This process is the analog of the uniform spanning tree in the presence of an arbitrary rank $n$ quenched gauge field in ${\rm U}(n)$ (or ${\rm O}(n)$, ${\rm SO}(n)$, ${\rm SU}(n)$, ${\rm Sp}(n)$), and generalizes the determinantal random cycle-rooted spanning forest which corresponds to the case of ${\rm U}(1)$ and ${\rm SU}(2)$. We describe the support of this process by means of a polymatroid which is described in terms of the circular and bicircular matroids. We also explain how the DLSF relates to the computation of currents in an electrical network in the presence of a gauge field. The partition function of the DLSF is shown to be the determinant of the twisted Laplacian, for which we also give a matrix-tree type theorem that computes its characteristic polynomial. We initiate the study of the twisted Laplacian determinant as a functional on the space of connections, and give a first order expansion of it in a neighborhood of the trivial connection, for which we provide a probabilistic interpretation. We extend the construction to infinite graphs, exhibiting the analogs of the classical wired and free distributions on spanning forests. We give a sufficient geometric condition on the local spectral gap for the existence of a Poincaré inequality as well as the uniqueness of the infinite volume measure, and the exponential decay of correlations. We show that on any recurrent graph, the wired and free DLSF coincide. We construct examples where they do not, for example, unlike in the classical case, on $\mathbb Z^d$ for $d\geq 3$.

We consider a random subgraph G_p(n,r,<s) of the complete distance graph G(n,r,<s) whose vertices are the r-element subsets of the set {1,...,n} and whose edges join pairs of subsets that intersect in fewer than s elements; each edge survives independently of the others with probability p. The stability of the Erdos-Ko-Rado theorem is the event that the independence number of the random subgraph equals that of the original graph, N = C_{n-s}^{r-s}. For fixed r and s, two-sided bounds for the critical probability, whose ratio depends on r and s, were obtained by P. A. Ogarok and A. M. Raigorodskii. In this paper we establish the order of the critical probability for growing r=r(n) and s=s(n): if min{C_r^s r^2, s r^3} = o(n), then the upper and lower bounds differ only by an absolute constant factor. The proof is elementary; only one of the two branches of the growth condition uses the complete nontrivial-intersection theorem of Ahlswede-Khachatrian.

We study the Solomon–Terao algebra and the Solomon–Terao complex of a central hyperplane arrangement $\mathcal A$ in $\mathbb C^\ell$. Let $\eta$ be a homogeneous polynomial of degree $d\geq2$ whose restriction to every flat of $\mathcal A$ has no critical point away from the origin. We prove that the Solomon–Terao algebra $\mathrm{ST}(\mathcal A,\eta)$ has top degree $|\mathcal A|+\ell(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{ST}(\mathcal A,\eta)$ is bounded below by that of the Orlik–Solomon algebra. Both results hold without assuming that the arrangement $\mathcal A$ is tame. Finally, we prove that $\mathcal A$ is tame if and only if all positive-degree homology groups of the Solomon–Terao complex associated with $\eta$ vanish.

In this paper, we study various structural properties like girth, chromatic num ber, etc. of the intersection hypergraph on the subgroups of nite non-cyclic abelian group and we characterize for the star and hypertree structures. Fur thermore, we characterize hypergraphs which are embeddable on plane, torus and projective plane.

Gromov hyperbolicity measures the geometric complexity of metric spaces and metric graphs through geodesic triangles and four-point conditions. While calculating the exact continuous hyperbolicity constant $\delta^*(G)$ of a graph $G$ typically requires evaluating quadruples of points over an infinite metric continuum, discretizing this domain is a fundamental challenge in metric graph theory. In this paper, we study the continuous Gromov hyperbolicity constant via the Gromov product. Our main result, the Quarter-Point Theorem, establishes that the search space for quadruples $(x,y,z,w)$ attaining the exact hyperbolicity constant $\delta^*(G)$ can be reduced from the entire metric graph to a finite or discrete subset $J'(G)$, comprising solely the vertices, edge midpoints, and quarter-points of $G$. This result holds for every graph, finite or infinite. This theoretical result provides a considerably simplified combinatorial framework for calculating hyperbolicity in complex networks and metric graphs. Furthermore, we exploit this discretization to significantly improve classical results by establishing and structural bounds relating the global hyperbolicity constant $\delta^*(G)$ with its restrictions to vertices ($\delta_V^*(G)$) and midpoints ($\delta_J^*(G)$).

We study the computational complexity of graph distortion distances, a family of graph metrics measuring how vertex distances differ between two graphs. Formally, these distances are defined in terms of standard matrix norms on the difference of distance matrices under an optimal vertex alignment. The resulting metrics have natural interpretations, corresponding to three ways of measuring the distortion of shortest-path distances: the maximum distortion among all vertex pairs (entrywise maximum norm), global aggregation (entrywise $p$-norms), and local aggregation (operator $p$-norms) of distortions. For maximum distortion, we show that approximation below a factor of two is NP-hard even for highly restricted trees. Due to the close connection to the Gromov-Hausdorff distance, we obtain a similar hardness result for the latter. In both cases, we show essentially optimal additive inapproximability. On the algorithmic side, we give a subcubic-time algorithm for optimal embeddings of connected graphs into paths, based on a structural decomposition along a diametral path. Additionally, we give a simple approximation algorithm when one input graph has a "relatively path-like" metric structure, i.e. has bounded pathlength. For local and global distortion, we derive various hardness results even for restricted inputs. Making use of a close connection between the global distortion distances and Gromov-Wasserstein distances, we give the first formal proof of NP-hardness of the latter, which has been claimed in the literature many times but has, to the best of our knowledge, never been correctly proven.

We study the number of improving steps that local search may require for $(2,2,r)$-MCA (Maximum Constraint Assignment), a bounded-occurrence version of MAX-CSP with valued constraints in which every constraint involves at most two variables, every variable occurs in at most two constraints, and every variable takes one of $r$ values. We consider the flip neighborhood, where two assignments are neighbors if they differ in the value of exactly one variable; moving to such a neighbor is called a flip. Local search repeatedly performs improving flips, that is, flips that strictly increase the objective value, until a local optimum is reached. Our main result is that $(2,2,5)$-MCA has the all-exp property: there is an infinite family of instances with initial assignments for which every sequence of improving flips ending in a local optimum has length exponential in the number of variables. Thus, on these instances, local search requires exponentially many steps regardless of which improving flip is chosen at each step. This is the first all-exp result for $(2,2,r)$-MCA for any value of $r$, and our result extends to every $r\ge 5$. In contrast, for $(2,2,2)$-MCA every sequence of improving flips has length at most quadratic in the number of variables~\cite{Kaz2020,KCJ2020}. We give a short, direct proof of this result.

Let $G$ be a finite abelian group of order $n$ and $X$ a finite $G$-set. We give a necessary and sufficient condition on $G$ for any $X$ to be determined up to isomorphism by certain combinatorial data. Specifically, if $M_H^X$ is the number of $H$-orbits in $X$, and $a_d^X$ is the number of $G$-orbits in $X$ of cardinality $d$, we characterize when equalities $M_H^{X_1} = M_H^{X_2}$ and $a_d^{X_1}=a_d^{X_2}$ for all $H \leq G$ and $d|n$, respectively, force an isomorphism $X_1 \simeq X_2$. Our main result is that this occurs if and only if $G$ is cyclic or $G$ is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to $L$-function factorization.

We study quotients of free associative algebras by the two-sided ideals generated by positive-degree symmetric, quasisymmetric, or free quasisymmetric polynomials. Free quasisymmetric functions give a word realization of the Malvenuto-Reutenauer Hopf algebra of permutations. For quotients on $n$ variables, finiteness in the symmetric and quasisymmetric cases follows from a theorem of Bergman–Isaacs when $n!$ is invertible in the field. Our main result is a nilpotence criterion for families of relations compatible with restriction to subalphabets. The criterion proves finiteness over every field for quasisymmetric, $r$-quasisymmetric, and free quasisymmetric coinvariants, with explicit degree bounds exponential in $n$ for fixed $r$. In the quasisymmetric and $r$-quasisymmetric cases, these improve the bounds inherited from Bergman–Isaacs. A principal application is to the free quasisymmetric case, where, for $n\geq3$, the group-invariant argument used for the symmetric case does not apply. For the symmetric quotient $C_n$, we prove that invertibility of $n!$ is also necessary for finiteness. In characteristic zero, an auxiliary presentation and the Golod–Shafarevich inequality give $\log\dim C_n\geq(1-o(1))n^5(\log n)^2$ as $n\to\infty$. The same inequality shows that quasisymmetric and free quasisymmetric coinvariant dimensions grow faster than every fixed power of $n!$. For all three families on several freely noncommuting alphabets, we determine all multigraded Hilbert series and, for symmetric coinvariants in characteristic zero, Frobenius characteristics from one-alphabet data. In characteristic zero, projection gives stable Schur multiplicities for compatible quotients in mixed commuting, anticommuting, and free alphabets.

We obtain very high genus asymptotics for double Hurwitz numbers when both the size $n$ and the genus $g$ go to infinity, under the constraint that $g$ grows much faster than $n$, more precisely that $\frac{2g}{n}-\ln n\to\infty$. Our proof relies on adapting arguments about the mixing time of the transposition random walk introduced by Diaconis and Shahshahani.

Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\ge3$ satisfies $|\mathcal{N}_S|\le p+1-\lceil p/(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv1\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

Bergeron, Garsia, Haiman, and Tesler conjectured that $(-1)^{n-\ell(\mu)}\nabla\widetilde H_\mu[X;0,t]$ and $\nabla\omega \widetilde H_\mu[X;0,t^{-1}]$ are Schur positive, and that $\nabla\omega \widetilde H_\mu[X;0,t^{-1}]-\nabla\omega \widetilde H_\nu[X;0,t^{-1}]$ is Schur positive when $\mu$ dominates $\nu$. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for $\nabla^r$ with $r\geq1$. The main new step is an identity: when $\mu$ covers $\nu$, the difference $\omega \widetilde H_\mu[X;0,t^{-1}]-\omega \widetilde H_\nu[X;0,t^{-1}]$ is minus a power of $t$ times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that $\nabla^r$ of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

Inspired by Steinitz's celebrated result (1913) and recent online vector-balancing results of Kulkarni et al. (2024) and Aden-Ali (2026), we introduce the online Steinitz problem: $T$ vectors in $\mathbb R^d$, each of $\ell_2$-norm at most $1$ and with total sum zero, arrive one at a time and must be assigned immediately and irrevocably to unused positions of an array. The goal is to keep every prefix sum of the resulting order small. We first obtain deterministic prefix-discrepancy bounds of $O(\log T)$ for $d=1$ and $O(\sqrt T)$ for $d\ge2$. Our main contribution is a general reduction that converts deterministic and randomized online vector-balancing algorithms, which assign signs to arriving vectors, into online ordering algorithms. Combined with recent vector-balancing results, this yields a randomized algorithm with prefix discrepancy $O(\log T+\sqrt{d\log T})$ and running time $O(dT\log T)$. The guarantee holds with high probability against an oblivious adversary. The reduction also gives bounds for sparse vectors and extends to inputs in arbitrary compact convex bodies. Finally, we prove matching lower bounds in both the deterministic and randomized cases.

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all nonempty sets $X\subseteq V(G)$ such that $N_{G}(X)\neq V(G)$. A graph $G$ is 1-binding if $b(G)\geq 1$. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with $\delta(G)\geq k$ that contain no $k$-factor for $k=1$, $2$. Subsequently, Tang and Zhang solved the problem for $k\geq 3$. Note that $\delta(G)\geq a$ is a necessary condition for the existence of an $[a,b]$-factor in a graph $G$. For $1\leq a<b$, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph $H_n^{a,b}$ among connected graphs with $\delta(G)\geq a$ that contain no $[a,b]$-factor. Note that $b(H_n^{a,b})<1$. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper determine the unique extremal graph $\mathcal{B}_n^{a,b}$ with the maximum spectral radius among 1-binding graphs of order $n$ with $\delta(G)\geq a$ that contain no $[a,b]$-factor. The toughness of a non-complete graph $G$ is $t(G)=\min\{\frac{|X|}{c(G-X)}\mid X\subseteq V(G)\, \text{and}\, c(G-X)\geq 2\}$. A graph $G$ is 1-tough if $c(G-X)\leq |X|$ for every $X\subseteq V(G)$ with $c(G-X)\geq 2$. Every 1-tough graph of order $n\geq2$ is 1-binding. Note that $t(\mathcal{B}_n^{a,b})<1$. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order $n$ with $\delta(G)\geq a$ containing no $[a,b]$-factor.