Papers

We prove an approximate analog of Vu's Theorem on economical bases of $k$th powers for non-integer powers. Fix a non-integer $\theta > 2$ and real $\tau > 0$. Let $s \geq \theta^2+9\theta^{3/2}+2 $ if $\theta > 3$ and $s \geq (\lfloor 2\theta \rfloor+1)(\lfloor 2\theta \rfloor+2)+1$ if $2 < \theta < 3$. We show the existence of a set $\mathfrak{X} \subseteq \mathbb{N}$ such that the number $R_{\mathfrak{X},s,\theta,\tau}(\Lambda)$ of integer solutions $(x_1,\ldots,x_s) \in \mathfrak{X}^s$ to the equation \begin{align*} |x_1^{\theta} +\cdots+ x_s^{\theta} - \Lambda| < \tau \end{align*} satisfies $R_{\mathfrak{X},s,\theta,\tau}(\Lambda) \asymp \tau \log(\Lambda)$ for all sufficiently large real $\Lambda > 0$.

Let $N(H)$ be the edge–vertex incidence matrix of a finite linear $s$-uniform hypergraph whose line graph is a cograph, and define \[ R_s(H)=(s+1)\operatorname{rank}_{\mathbb R}N(H)-s|E(H)|. \] The inequality $R_s(H)\ge0$ holds, with equality characterized by affine planes of order $s$ on every row-dependent component. Under the corresponding finite-net completion hypothesis, connected systems with bounded $R_s$ admit an explicit classification. If $F$ is connected, $R_s(F)=d\le s-1$, the completion hypothesis holds through deficiency $d$, and $s>(d-1)^2$, then either $N(F)$ has full row rank and $|E(F)|=d$, or $F$ lies in a unique affine plane of order $s$. In the latter case, for some $0\le h\le d$, the system contains every line in $s+1-h$ parallel classes and exactly $d-h$ additional lines from the remaining $h$ classes. Conversely, each such system has defect $d$. Bruck's theorem gives the classification for every fixed $d$ and all sufficiently large $s$, while Metsch's completion theorem covers $d=o(s^{1/3})$. As a global consequence, if the total defect is at most $D$ in the completion range, deleting at most $D$ edges leaves a vertex-disjoint union of finite nets and affine planes. The associated column defect and row leverage are determined explicitly as well.

We study morphisms from symmetric powers of exterior powers to determinant-twisted endomorphism representations of general linear groups. At the minimal positive determinant twist, we prove vanishing over every field of characteristic different from two: if the exterior degree r >= 3 is odd, the symmetric degree satisfies d >= 3, and the underlying space has dimension rd, then the corresponding equivariant Hom space is zero. The proof uses a block-exchange sign and a universal root-subgroup identity, so it also applies in small odd characteristics without semisimplicity. We then determine the second-twist spaces for trivectors in characteristic zero. In dimension 3m and degree 2m, they are scalar and one-dimensional for m = 2, and zero for m >= 3. Plethystic conjugation reduces the latter vanishing to an elementary weight-support bound for exterior powers of the ten-dimensional space of ternary cubics. Finally, over fields of characteristic zero or odd characteristic, an exact computer-assisted classification in dimension nine gives a one-dimensional scalar Hom space in characteristic five and zero in characteristic zero and in every odd characteristic other than five.

We evaluate the series $\sum_{n\ge1} z^n\big/\!\big(n^2\binom{4n}{n}\big)$, equal to $-\tfrac{z}{4}\,{}_5F_4\!\left(1,1,1,\tfrac43,\tfrac53;\tfrac54,\tfrac32,\tfrac74,2;\tfrac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $\theta=j\pi/N$. Each value equals $c\,\pi^2$ plus a universal rational quadratic form in three logarithms, with $c=-\tfrac13\left(1-\tfrac{2j}{N}\right)^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.

Let $0\leq\mu_1\leq\cdots\leq\mu_n$ and let $\lambda>\sum_{i=1}^n\mu_i$. Let $X_1,...,X_n$ be independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$, and write $D_i := \lambda-\sum_{k=1}^{i-1}\mu_k$ for $1\leq i\leq n$. We prove that $$ \inf_{X_1,...,X_n} \mathbb{P}\left( \sum_{i=1}^nX_i<\lambda \right) = \min_{1\leq i\leq n} \prod_{j=i}^n \left( 1-\frac{\mu_j}{D_i} \right). $$ The bound is sharp and is attained. This proves Samuels' conjecture. Feige's conjecture is thereby resolved, since it follows immediately from the equal-means case. The proof is self-contained.

We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid $H$ satisfying $1<\rho(H)<\rho(G)$, where $\rho$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy–Davenport theorem: we define a parameter $\mu(G)$ and prove that $\rho(XY)\geq \min\{\mu(G),\rho(X)+\rho(Y)-1\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $\rho(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.

Write $\mu_t$ for the $t$-th roots of unity and $z^{\pm1}$ for $r$ free reciprocal pairs. We study $\Phi_{t,r}(\beta)=s_{\lambda}(\mu_t,z^{\pm1})$, $\beta=\lambda+\delta$, and the question the companion paper left open after $r=1$: when does it vanish? We factor the evaluation into classical branching followed by a torsion filter, and the shape depends on the parity of $t$: the point lies in the orthogonal group with determinant $(-1)^{t+1}$. For odd $t$ it sits in the identity component: an ordinary restriction $SO_{2R'+1}\downarrow SO_{2m'+1}\times SO_{2r}$, the filter an odd orthogonal character at a principal element of order $h+1$. For even $t$ in the other: a twining, a virtual expansion, and a torsion element regular but not principal; there we prove the filter, with its sign. One description covers both: the filter is nonzero exactly when the shifted point is regular semisimple. Both are minimal-level fusion projections: the even of type $C$, the odd's tensor sector of type $B$. Affine folding accounts for $0,\pm1$; what it does not survives as conjectures. The highest surviving weight is the dominant vertex of the numerator's Newton polytope minus the denominator's — the latter proved, the former conditional on a single-orbit property — and the class there is conjecturally primitive, $\pm$ the generator of the rank-one quotient. For odd $t$ and one $\Lambda$ that numerator is a signed transversal count in $\{0,\pm1\}$ by the equal-rank character formula, leaving one division. We invert it in closed form, along an arithmetic progression; the quotient is $\pm\epsilon_t\det M$ for an explicit $0/{\pm}1$ matrix — an interval matrix up to signs, hence totally unimodular, which settles (L1). The fibre count is a permanent, odd only when $1$, so at a dominant index a multi-hit fibre sums to zero. Two extremal statements remain. What is unproved is measured, in both parities.

Let $p_k(n)$ be the number of $k$-polymatroids on $[n]$. We show that for every fixed $k \geq 1$, we have \[ \left\lfloor \frac{k}{2} \right\rfloor \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)) \le \log_2 p_k(n) \le k \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)). \] We also show that for $k \geq 2$, almost all $k$-polymatroids are (i) connected, (ii) proper, and (iii) not linearly representable over any field.

This paper presents new theoretical results on generalizing the Jaccard distance for lattices and real valuations. We demonstrate that when the valuation is strictly positive, monotone, and modular, the Jaccard distance satisfies the triangle inequality on arbitrary lattices, effectively generalizing earlier results that depended heavily on distributivity. Moving to relatively complemented distributive lattices (which safely drop the requirement for the global bounds found in Boolean algebras), we prove the triangle inequality holds as long as the valuation is positive, monotone, supermodular, and $\log$-submodular. Additionally, we adapt the symmetric-difference Jaccard formulation for submodular valuations to sectionally complemented distributive lattices. Shifting to necessary conditions, we prove that supermodularity is a strict requirement for the standard generalized Jaccard distance to operate as a valid metric. Finally, we map the practical value of relaxing these structural constraints to computational fields like quantum information theory, formal concept analysis, and machine learning, closing with a brief look at open mathematical problems.

Near maximum distance separable (NMDS) codes of dimension 3 and length n over the finite field with q elements are equivalent to (n,3)-arcs in PG(2,q). For every odd prime power q we construct, by adding five suitable points to a conic of PG(2,q), a family of [q+6,3,q+3] NMDS codes and determine their weight distributions completely; three distinct weight enumerators occur, governed by two explicit quadratic-character conditions on the parameters. For every odd prime power q, no code of our family is monomially equivalent to a code of the recent [q+6,3,q+3] NMDS family of Fan, Wang and Xu, even where the weight enumerators of the two families coincide: the separating invariant is a triple of geometric data attached to the underlying arc. Extending the configuration by a sixth point on a distinguished external line, we further obtain [q+7,3,q+4] NMDS codes, together with their weight distributions, for every odd prime power q >= 11 (admissible parameters exist for no q <= 9); the existence proof combines exact and Weil-type character sum estimates with a finite computer verification. Our proofs are purely geometric and rest on a simple counting identity for the trisecant lines of a point set obtained by extending a conic. All the codes constructed are optimal locally recoverable codes with locality 2.

Stanley's tree-isomorphism conjecture asks whether the chromatic symmetric function distinguishes nonisomorphic trees. We give a reconstruction criterion that permits repeated leaf-component orders. For a proper tree, collapse each leaf component to its center and record its order as a vertex weight. We prove that the chromatic symmetric function reconstructs the tree whenever every nonleaf vertex of this weighted core has a weight that occurs nowhere else in the core. Repetitions among core leaves are unrestricted. The proof uses only the leading star-basis partition and the coefficients immediately above it. As a consequence, the conjecture holds for an infinite class of diameter-six trees not covered by the condition that all leaf-component orders are distinct. We also give a canonical integer-partition model for arbitrary diameter-six trees and an exact cut-partition implementation intended for further work. The unrestricted diameter-six case remains open.

Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six – the first dimension in which several continuous families coexist with an isolated solution – has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.

We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient–submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander–Reiten quiver, extending the classification of Oppermann–Reiten–Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.

Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology of the associated toric variety when the polytope is rational. Maxim-Schuermann computed the intersection cohomology signature of a projective toric variety, corresponding to the case of a polytope with rational vertices. Using the combinatorial framework of BBFK, we show that the Maxim-Schuermann formula extends to arbitrary convex polytopes. Finally, we discuss a version of the Hodge index theorem for polytopes.

Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé in 2010. In this paper, we consider a rainbow analogue of Lehel's conjecture in the setting of properly edge-coloured complete graphs. We prove that, for sufficiently large n, every properly edge-coloured Kn admits a partition of its vertex set into two vertex-disjoint rainbow cycles

We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing that, in this setup, a Stokes filtration is fully determined by the data of either its recessive or subdominant solution spaces, and making the link with results of T. Mochizuki, we show that the Fourier transform amounts to exchanging recessive and subdominant solutions via the Gale transform of configurations of points in projective spaces. (2) We show that the equivalence between recessive solutions and Stokes local systems is deeply connected with the triality relating point configurations, superperiodic linear difference equations and frieze patterns obtained by Morier-Genoud-Ovsienko-Schwartz-Tabachnikov: Up to signs, the coefficients of the difference equations and friezes coincide with the nontrivial Stokes matrix entries. It follows from this Stokes-frieze correspondence that the Fourier transform of Stokes representations is given by their combinatorial Gale transform, leading to explicit closed formulas.

The cluster graphs on $n$ vertices, the disjoint unions of complete graphs, have the integer partitions of $n$ as their isomorphism classes, and the quotient edit distance $q^*(\lambda,\mu)=\min_{\sigma\in S_n}|E(G_\lambda)\triangle\sigma E(G_\mu)|$ makes that set a metric space. Its geometry and its complexity both issue from one identity: $q^*$ is an affine function of the maximum of $\lVert X\rVert_F^2$ over the contingency tables with margins $\lambda$ and $\mu$. Our main result is an explicit optimal embedding. The weighted dyadic sums of the Ferrers staircase, taken at the critical exponent $\frac14$, give a map $F_n$ into $\ell_2^{\,<4n}$ that acts on a single partition and is computable in $O(n)$ time, and its distortion is $\Theta(n^{1/4})$. That order is optimal, since $c_2(\mathcal K_n)=\Theta(n^{1/4})$: the lower half follows from a $\Theta(\sqrt n)$-dimensional Hamming cube of partitions and Enflo's theorem, so the determination needs no other external input. The analytic core is a scale-free inverse inequality for every integer sequence with $v(1)=v(N+1)=0$ and $v(s)-v(s+1)\in s\mathbb Z$: its critical dyadic energy is at least $\lVert v\rVert_1^2/(63504\sqrt{\mathrm{TV}(v)})$. Combinatorially the same identity yields two explicit $\ell_1$ models, the vertex-mass metric on sorted degree sequences with $\frac12\delta_1\le q^*<\frac32\delta_1$ and the block-energy metric with $q^*\le B\le2q^*-1$, both constants optimal; hence $c_1(\mathcal K_n)\le2$, and an $O(n\log n)$-time algorithm returns an alignment of cost below $2q^*$ carrying the certificate $q^*\in[\lceil(B+1)/2\rceil,B]$. Computationally, deciding $q^*(\lambda,\mu)\le Q$ is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable. The best constant in the inverse inequality remains open; an exactly solvable chirp family caps it at $\frac23$.

The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse $p$-random setting, provided $p$ is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability $p$, and the probability of success.

Christ [Chr01] looked at a certain boundedness problem for trilinear operators. In this paper we set up a framework of mixed-norm Brascamp-Lieb inequalities and identify a new interesting regime not covered by the study of classical Brascamp-Lieb. Positive results in [Chr01] can be viewed as the first nontrivial progress in this new regime. We will also prove another mixed-norm Brascamp-Lieb inequality, showcasing how one can use a tensor product trick to slightly sharpen Christ's argument and also obtain the endpoint case. We then discuss the connections between mixed-norm Brascamp-Lieb and the Kakeya Conjectures, as well as unique new difficulties for these mixed-norm Brascamp-Lieb inequalities compared to the classical setting.

Let k and l be two non-negative integers with k > l. A graph G is (k,l)-stable if alpha(G - S) >= alpha(G) - l for every subset S of V(G) with |S| = k, where alpha(G) denotes the independence number of G. Dong and Wu established that alpha(G) <= floor((n - k + 1)/2) + l for a (k, l)-stable graph G, where n is the order of G. A (k, l)-stable graph G is tight if alpha(G) = floor((n - k + 1)/2) + l. In this paper, we provide a complete characterization of tight (k, 0)-stable graphs for k >= 4. In particular, we prove that tight (k, 0)-stable graphs are K_{k+1} and K_{k+2} for k >= 5, which not only extends the result of Liu, Song and Wang [J. Graph Theory 110(2) (2025), 193-199] from k >= 24 to k >= 5, but also proves the conjecture of Dong and Luo [Electron. J. Comb. 32(4) (2025), 4-45] once more.

Given a graph $G$, we show that the mixed metric dimension of $G$ is exactly $\ell(G)+2c(G)$ if and only if $G$ is either a cactus graph in which every cycle has precisely one vertex of degree at least $3$, or a balanced $\Theta$-graph, where $\ell(G)$ and $c(G)$ denote the number of leaves and the cyclomatic number of $G$, respectively. This provides an affirmative answer to a conjecture proposed by Sedlar and Škrekovski (2021).

An edge-weighted graph $H$, possibly with loops, is antiferromagnetic if its adjacency matrix is entrywise nonnegative and has at most one positive eigenvalue, counted with multiplicity. We show that, for any graph $G$ with $d_v:=\operatorname{deg}_G(v)$, $$\operatorname{hom}(G,H) \ge \prod_{v\in V(G)} \operatorname{hom}(K_{d_v+1},H)^{\frac{1}{d_v+1}},$$ whenever $H$ is antiferromagnetic. In fact, we prove a vertex-inhomogeneous strengthening of this inequality, allowing a different fugacity vector at each vertex of $G$. This gives a common generalization of the lower-bound inequalities of Sah, Sawhney, Stoner, and Zhao for independent sets, of Csikvári for $q$-colorings, and of the authors for semiproper colorings with at most two proper colors. Furthermore, it confirms recent conjectures of the authors and of Davies and LeBlanc. A key ingredient, of independent interest, is a strengthening of the delete-one form of Shearer's inequality for Lorentzian measures, which provides a new approach to graph homomorphism inequalities.

Let G be a cubic graph of order n. The induced cycles vertex number cind(G) is the largest size of a vertex set that induces a 2-regular subgraph of G. By gamma_1,2(G) we denote the (1,2)-domination number of G. Erves and Tepeh introduced the trilobite graphs T_n, which satisfy gamma_1,2(T_n) > cind(T_n). They stated two conjectures, the first of which says that every cubic graph G with cind(G) >= n/2 + 2 satisfies gamma_1,2(G) <= cind(G). The second says that a connected cubic graph G satisfies gamma_1,2(G) > cind(G) if and only if G is a trilobite. We show that both conjectures are false. A computer search finds counterexamples that are not trilobites already for n = 18, 20 and 22. We also construct an infinite family H(k) of order n = 20 + 4k. For every k >= 0 we prove that cind(H(k)) = n/2 + 2 and gamma_1,2(H(k)) = n/2 + 3. Hence both conjectures fail for infinitely many orders n.

Let Phi be an irreducible reduced crystallographic root system of rank r, and let N be the number of positive coroots. A previous theorem (arXiv:2608.16363, Theorem 3.3) identifies the first pole below 2/h as q_2 = (r-1)/(N-1) and expresses its residue as a sum of periods attached to the simple walls of the dominant chamber. We evaluate that wall-period sum for all four classical families. The identity q_2 (N-1) = r-1 removes the radial variable from every wall integral. The resulting projective integrals reduce to mixed Dotsenko-Fateev chambers in type A_r, to those chambers together with a Selberg endpoint in types B_r and C_r, and to a single chamber family in type D_r. The chamber recurrences give explicit sine weights in types A, B, and C. In type D, a terminating basic-hypergeometric sum at a root of unity reduces to a finite cyclotomic product. We also show that the corresponding exceptional wall restrictions are not reflection arrangements, so this particular reduction does not extend directly to the exceptional types.

In any fixed dimension, consider a finite collection of sets, each consisting of exactly two disjoint, closed, convex pieces. We show that to guarantee the intersection of the whole collection also consists of exactly two such convex pieces, it suffices to verify the same two-piece structure for all intersections of some subfamilies of intermediate size, thus answering a question of Gil Kalai.

Let $\mathcal{P}_{n,d}$ be the space of polynomials in $n$ variables over $\mathbb{F}_q$ of degree at most $d$. Two polynomials $f,g\in\mathcal{P}_{n,d}$ intersect if $f(\mathbf a)=g(\mathbf a)$ for some $\mathbf a\in\mathbb{F}_q^n$. A star consists of all polynomials $f\in\mathcal{P}_{n,d}$ satisfying $f(\mathbf a)=b$ for fixed $\mathbf a\in\mathbb{F}_q^n$ and $b\in\mathbb{F}_q$. We completely classify the maximum intersecting families in $\mathcal{P}_{n,d}$. When $n=1$ and $d\geq 2$, it was previously shown that all maximum intersecting families are stars. We prove that the same conclusion holds for all $n\geq 2$ and $d\geq 2$ when $q$ is odd. When $q$ is even, however, the situation is more subtle, and a new phenomenon emerges: for $q\geq 4$, maximum non-star examples exist precisely when $d\leq n$. Along the way, we prove two further results of independent interest. First, we determine the span of nonvanishing polynomials in $\mathcal{P}_{n,d}$. Second, we characterize all linear functionals $\Psi\colon\mathcal{P}_{n,d}\to\mathbb{F}_q$ whose kernels are disjoint from the set of nonvanishing polynomials. The first result plays a crucial role in the proof of our main result; the second is a Gleason–Kahane–Żelazko theorem for polynomials of bounded degree over finite fields.

Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.

Let $J_G$ denote the binomial edge ideal of a finite graph $G$ in the polynomial ring $S$. We determine all triples $(n,t,d)$ with $n\geq 3$ for which there exists a finite connected graph $G$ on $n$ vertices with ${\mathrm{depth}}(S/J_{G})=t$ and $\dim(S/J_G)=d$.

For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In his 1977 formulation of the problem, Erdős discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erdős's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.

Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper–Hanna–Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.