Papers

In this paper we investigate the existence of a maximal-cardinality \(t\)-resolvable quantum Latin square (\(t\text{-}\text{RQLS}(v)\)). We determine the existence of a maximal-cardinality \(1\text{-}\text{RQLS}(v)\) with \(17\) possible exceptions based on the established research result and construct a maximal-cardinality \(2\text{-}\text{RQLS}(n^2)\) for every integer \(n\geq6\).

A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of the edge set $E(G)$ into $j$ matchings and $k$ induced matchings. Hocquard, Lajou, and Lu\v zar found a subcubic planar graph of girth $3$ that has no $(1,2^5)$-packing edge-coloring and also conjectured that every subcubic planar graph has a $(1,2^6)$-packing edge-coloring. We also notice that for every fixed positive integer $k$ there exists a subcubic planar graph with girth $k$ that is not $(1,2^3)$-packing edge-colorable. It is natural to consider what is the minimum positive integer $k_1$ such that every subcubic planar graph with girth at least $k_1$ is $(1,2^5)$-packing edge-colorable. Furthermore, we also consider what is the minimum positive integer $k_2$ such that every subcubic planar graph with girth at least $k_2$ is $(1,2^4)$-packing edge-colorable. In this paper, we show both $k_1$ and $k_2$ are finite, and in fact $5 \le k_1 \le 12$ and $6 \le k_2 \le 16$.

We study Fulton's universal Schubert polynomials $\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $\Delta_1\times\cdots\times\Delta_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirković–Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\mathfrak S_w(c)$ implies $\Lambda_Q$-rigidity of $Z_{\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\in S_{10}$ for which the three geometric basis elements are distinct.

A digraph is supereulerian if it contains a spanning closed trail. Let $\alpha_2(D)$ denote the maximum cardinality of a vertex set inducing no 2-cycle. In this paper, we characterize supereulerianity in a strong digraph $D$ with $\alpha_2(D)=4$ by proving that a strong digraph $D$ with $\alpha_2(D)=4$ and $\lambda(D)\ge 2$ is supereulerian if and only if $D$ does not belong to an exceptional family $\mathcal H$ of $2$-arc-strong digraphs with $\alpha_2(D)=4$. Furthermore, every strong digraph satisfying $\alpha_2(D)=4$ and $\lambda(D)\geq3$ is supereulerian.

We develop a few calculus rules to compute or lower bound the facial distance of a polytope. We illustrate our calculus rules on various popular polytopes. In particular, we provide a provide a lower bound on the facial distance of the Birkhoff polytope.

Let $\lambda_1(G) \geq \dots \geq \lambda_n(G)$ be the adjacency spectrum of a graph $G$ on $n$ vertices. The spectral distance $\sigma(G,H)$ between $n$-vertex graphs $G$ and $H$ is the Manhattan distance between their spectra, i.e. $\sigma(G,H) = \sum_{i=1}^n |\lambda_i(G) - \lambda_i(H)|$. Given a set $\mathcal{G}$ of pairwise non-isomorphic graphs of order $n$, the spectral diameter of $\mathcal{G}$ is defined as $\mathrm{sdiam}(\mathcal{G}) = \max\{\mathrm{secc}_{\mathcal{G}}(G) : G \in \mathcal{G}\}$, where $\mathrm{secc}_{\mathcal{G}}(G) = \max\{\sigma(G,H) : H \in \mathcal{G}\}$ is the spectral eccentricity of $G \in \mathcal{G}$. Among six conjectures on spectral distances posed by Z. Stanić in 2012, two conjectures related to the spectral diameter of certain graph classes remained open. One of them concerns the spectral diameter of the set $\mathcal{B}_n$ of all connected bipartite graphs of order $n$, while the other, of the set $\mathcal{T}_n$ of all trees of order $n$. More precisely, Stanić conjectured that $\mathrm{sdiam}(\mathcal{T}_n) = \sigma(P_n, K_{1,n-1})$, where $P_n$ is the path graph, while $K_{1,n-1}$ is the star, and that $\mathrm{sdiam}(\mathcal{B}_n) = \mathrm{secc}_{\mathcal{B}_n}(K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor})$, where $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$ is the complete bipartite graph. In this paper, both of these conjectures are disproved.

For a connected graph $G$, let $\gamma(G)$ and $\gamma_c(G)$ denote its domination number and connected domination number, respectively. Let $M(n,k)$ be the maximum of $\gamma_c(G)-\gamma(G)$ over all connected $n$-vertex graphs of minimum degree at least $k$. Alon proved that \[ 2\left\lfloor\frac{n}{k+1}\right\rfloor-O(1)\le M(n,k)<\frac{n}{k+1} \bigl(\log\lceil\log(k+1)\rceil+3\bigr). \] He proposed a problem to determine or estimate $M(n,k)$ for $n-1\ge k\ge 3$, and particularly remarked that it would be interesting to close the $\log\log(k+1)$ gap between the upper and lower bounds and decide whether or not $M(n,k)=\Theta\bigl(\frac{n}{k+1}\bigr)$. We give an asymptotic answer to Alon's problem for sufficiently large $k$. More precisely, $M(k+1,k)=0$. For $n>k+1$, let \[ \nu=\frac{n}{k+1} \text{ and } \Phi(x)= \frac{1}{x\log\!\frac{x}{x-1}} \ (\text{for }x>1). \] As $k\to\infty$, uniformly over all integers $n>k+1$, \[ M(n,k)= \big(\Phi(\nu)+o(1)\big) \nu\log\log(k+1). \]

An equitable $k$-coloring of a graph partitions its vertex set into $k$ independent sets whose sizes differ by at most one; the least such $k$ is the equitable chromatic number $\chie(G)$. Every known bound on $\chie$ valid for all graphs, beginning with the Hajnal–Szemerédi theorem, is linear in the maximum degree $\Delta$, and the star $K_{1,\Delta}$, for which $\chie=\ceil{\Delta/2}+1$, shows that no general bound below $\Delta/2$ exists. We prove that this obstruction is a shortage of vertices rather than an effect of the degree: every graph with $|V(G)|\ge3\chi(G)\Delta$ satisfies $\chie(G)=O\bigl(\chi(G)^{3/2}\sqrt{\Delta/\ln\Delta}\bigr)$ throughout the range $\chi(G)\le(\Delta/\ln\Delta)^{1/3}$, so that for graphs of large order the degree enters only through $\sqrt{\Delta/\ln\Delta}$, with the chromatic number governing the rest. For each fixed $\ell$, $\ell$-colorable graphs of sufficiently large order satisfy $\chie\le\bigl(2\sqrt2\,\ell\sqrt{\ell-1}+o(1)\bigr)\sqrt{\Delta/\ln\Delta}$, while a probabilistic construction supplies graphs of arbitrarily large order, bipartite when $\ell=2$, with $\chie\ge\tfrac13\sqrt{(\ell-1)\Delta/\ln\Delta}$: the order of growth $\Theta\bigl(\sqrt{\Delta/\ln\Delta}\bigr)$ is exact for every fixed chromatic number, and the extremal constant is determined up to a factor $O(\chi(G))$. All upper bounds are constructive, and a prescribed-anchor variant of the construction produces equitable colorings of bipartite graphs with $O(\sqrt\Delta)$ colors in optimal linear time.

The enumeration of chemical structures traditionally relies on graph- and group-theoretic techniques, including Pólya theory, in order to account for stereochemical symmetries. These methods require bespoke derivations and methods for each molecular family of interest, and quickly become very intricate for nontrivial classes. In this work, we demonstrate that analytic combinatorics provides a unifying, highly extensible, and relatively simple framework for enumeration problems. Using the symbolic method, we show that classical counting problems, such as the enumeration of acyclic alkanes and other hydrocarbon families, can be naturally solved via concise combinatorial specifications of the class of interest. We first demonstrate the simplicity of this approach by replicating existing results in this framework, requiring significantly less work than previously necessary, and then extend these ideas to more complex problems, counting broader classes of molecules accounting for both tetrahedral and E/Z stereoisomerism, counting the distribution of stereogenic units, and enumerating achiral and meso compounds. All results are made reproducible through CombOL, our open-source software package which automates the translation of combinatorial specifications into generating functions using the symbolic method, and performs both univariate and multivariate enumeration.

Braess' Paradox is the phenomenon in which adding an edge to a congestion network increases total travel time. We study the paradox in directed rectangular grids where every edge shares the latency function $\ell(x)=ax+b$ and an added chord has latency $\ell_{*}(x) = cx+d$, where $a > 0$ and $b,c,d \ge 0$. Using an analogy with electrical networks, we bound the change in total travel time. We then give a necessary and sufficient condition for a chord to induce the paradox for some choice of nonnegative coefficients and compute the exact proportion of such chords in all grids with dimensions at most $100$. Such chords are scarce, and the fraction is maximized near an aspect ratio of $2:1$. We then improve the established $4/3$ upper bound on the Braess Ratio to one depending only on the grid dimensions, approaching $1.207$ on squares and $4/3$ on thin grids. Finally, we prove any ratio-maximizing chord must have zero latency.

We study when a polynomial sequence $\{p_n(x)\}$ satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences $p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$, we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order $\ell \geq 3$. We then develop a production-matrix framework for arbitrary orthogonal polynomial sequences (OPS), not restricted to the Riordan-array-type case: for any lower-triangular invertible coefficient matrix $A$ of an OPS, the production matrix $P_{A^{-1}}=ASA^{-1}$ is always the tridiagonal Jacobi matrix encoding the three-term recurrence, and the first column of $A^{-1}$ always gives the moment sequence – even when, as for the Legendre polynomials, the recurrence coefficients are non-constant and no Riordan array representation of the coefficient matrix exists. We illustrate this with Legendre and Chebyshev polynomials as, respectively, non-Riordan-type and Riordan-type examples, and give an explicit closed-form coefficient matrix and its inverse for the generalized Gegenbauer–Humbert OPS. Finally, we extend the framework to $d$-orthogonal polynomial sequences, showing that the coefficient matrix of a $d$-orthogonal sequence is always invertible, that its associated production matrix is $(d+2)$-banded (lower Hessenberg) and encodes the corresponding $(d+2)$-term recurrence, and that the first $d$ columns of the inverse matrix recover the moment sequences of the defining vector functional – identifying $d$-orthogonality with $(d+1)$-Hessenberg generalized Riordan arrays and extending the classical $d=1$ tridiagonal correspondence.

We prove that a positive-definite quadratic form has nonnegative conorms if and only if it is matroidal, resolving a conjecture of Dutour Sikirić and Kummer. We show that the positive conorm support determines a regular matroid, and the positive conorms are the coefficients of a unimodular integral decomposition of the form. As a result, we show that a positive-definite form lies in the tropical Schottky locus if and only if its conorms are nonnegative and its positive conorm support matroid is cographic. This answers a question of Chua, Kummer, and Sturmfels and provides algorithms for the tropical Schottky decision and recovery problems in every genus.

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.

We show that the literal arbitrary-packet form of the refined-decoupling estimate printed in the Guth–Iosevich-Ou-Wang proof of the planar pinned Falconer theorem is false, even after its tube dimensions are normalized. An explicit collar construction gives a fixed-power counterexample: all packets are active on one square while none of their smaller labelled tubes meets that square. We then give a non circular repair of the original proof route. The analytic input is an enlargement-stable arbitrary-packet theorem in which a packet is concentrated on an $a$-dilate and multiplicity is counted with a strictly larger $b$-dilate. We prove this theorem directly from weighted $\ell^2$ decoupling by an induction that tracks the dilation margin through parabolic rescaling. The corrected theorem applies directly to the original Falconer parent packets after an exact frequency truncation and an absolute small-packet cutoff; no parent-to-canonical decomposition is needed. We then rebuild the good-tube incidence estimate, retain the neighborhood forced by local constancy, and supply a uniform regularization and limiting argument. The principal frequency exponent remains $-(\alpha+1)/3$, so the energy argument closes exactly for $\alpha>5/4$. The pinned theorem itself is not contradicted and is also known through later microlocal methods. A later canonical wave-packet treatment of refined decoupling overlaps with the activity-tube viewpoint but not with the counterexample or the repaired Falconer proof chain.

We prove a formula for the linear coefficient of the Ehrhart polynomial in terms of the Tutte polynomial. Furthermore, we characterize the matroids that extremize this Ehrhart coefficient, proving a conjecture of Ferroni.

We prove that the six-vertex graph with edge set $\{ab,bc,cd,de,af,bf,df\}$ has the Erdős-Hajnal property. The proof adapts the iterative-sparsification method of Nguyen, Scott, and Seymour within the comb-based framework of Huang, Ju, and Zhou.

In this work, we present the first analysis of low-degree polynomial threshold functions for the natural hypothesis testing problem of detecting the noisy random lift of a base $d$-regular graph from a uniformly random $d$-regular graph. Along the way, we obtain a new result for the distribution of short cycle counts in noisy random lift up to logarithmic lengths, which generalizes results by McKay, Wormald, and Wysocka and by Johnson in the case of random regular graphs, and results by Greenhill, Janson, and Ruciński and by Fortin and Rudinsky in the case of random lifts.

Bang (1976) proved the inequality for matrix permanents $\mathrm{per}^2(A) \geq 2^{-2n}\mathrm{per}(A \otimes J_2)$, where $J_2$ is the $2 \times 2$ all-ones matrix and $A$ is any $n \times n$ matrix with non-negative entries. We show that, if $A$ is any $n \times n$ real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant $2^{-2n}$ by $1 / \binom{2n}{n} = (n!)^2 / (2n)! > 2^{-2n}$. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if $p$ and $q$ are monic real-rooted polynomials of degree $n$, then $(p \boxplus_n q)(x)^2 \geq (p^2 \boxplus_{2n} q^2)(x)$ and $(p \boxtimes_n q)(x)^2 \geq (p^2 \boxtimes_{2n} q^2)(x)$ for all $x \in \mathbb{R}$, for $\boxplus_n$ and $\boxtimes_n$ the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree $n$.

We define an invariant of alternating links—a homogeneous, four-variable Laurent polynomial—that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $\Omega(\sqrt{\log n})$ should hold.

The singular difference graph, denoted by $\Gamma$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $\Gamma$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $\Gamma$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $\Gamma$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tilde\Omega(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

We use tools and techniques from $p$-adic analysis and algebraic number theory to study the algebraicity of the hard square entropy constant and its high dimensional analogues. Specifically, we study arithmetic properties of $a_d(n)$, the number of independent sets in the $d$-dimensional discrete torus, and the associated entropy constants $\kappa_d=\lim_{n\to\infty}a_d(n)^{1/n^d}$. It is not known whether $\kappa_d$ is algebraic or transcendental for $d>1$. Using the fact that the sequence $a_d(p^k)$ converges $p$-adically for every prime $p$ and other arithmetic facts, we present a collection of criteria for the algebraicity of $\kappa_d$ and bound the number of possible values of prime powers $p^k$ for which $a_d(p^k)=\kappa_d^{p^{kd}}$.

We study regular unimodular triangulations, the integer decomposition property, and Ehrhart-theoretic properties of two families of Hermite normal form simplices. We first consider the one-row case associated with the vector $(N - 1, \dots ,N - 1 , N)\in \mathbb{N}^d$, and completely characterize when the corresponding simplices admit a regular unimodular triangulation. Our constructions are explicit and also yield closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial. Moreover, we prove Ehrhart positivity and derive explicit dimension-dependent conditions under which the Ehrhart polynomial is not unimodal. Finally, we extend our approach to the two-row cases associated with $(1, \dots ,1 , N)\in\mathbb{N}^d$ and $(M-1, \dots ,M-1, M, 0)\in\mathbb{N}^d$. In these cases, we construct regular unimodular triangulations, derive closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial, and prove Ehrhart positivity.

In this paper we study irreducible simplicial arrangements of projective planes in $\mathbb{P}^{3}(\mathbb{R})$ from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Grünbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types $A_4$ and $B_4$ admit a special vertex, whereas those of types $D_4$ and $F_4$ do not. A simplicial deletion chain inside $B_4$ supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Grünbaum-Shephard defects for these arrangements.

A $p$-ary bent partition of $\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\mid p^n$, so $K=p^t$; nonempty cells further give $1\le t<n$. In even dimension, the classical cell-size theorem yields $K\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\le\lfloor n/2\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.

We construct a new family of permutation group polynomials over finite fields of odd characteristic and explicitly provide its companion. Moreover, we precisely determine the number of permutation group polynomials of this form and those that are equivalent to this new family. In addition, we completely solve the problem of enumerating permutation group polynomials of the various forms presented in Hasan and Kumar (2026), as well as permutation group polynomials which are equivalent to these families.

We compute the diagonal $F$-threshold of binomial edge ideals associated to graphs in the case of cycles, cycles with whiskers, bipartite graphs, and block graphs. Moreover, we provide general bounds for any graph.

The canonical Bethe approximation gives a deterministic approximation to the permanent of every nonnegative matrix within a factor of $(\sqrt{2})^n$. We improve the base of this exponential factor: for some absolute constant $c<\sqrt{2}$, there is a deterministic polynomial-time $c^n$-approximation for the permanent of every nonnegative matrix. This shows that the canonical Bethe guarantee is not a barrier for deterministic approximation of the permanent. The proof augments the Bethe lower bound with a new certificate tailored to matrices on which that lower bound loses nearly the full factor. The author supplied the high-level plan of attack, and the proof was developed in an interaction with ChatGPT 5.6 Sol Pro. The author subsequently verified the results. Codex assisted with proof checking, manuscript assembly, and typesetting.

Let $V$ be an $n$-dimensional vector space over a finite field of order $q$. Let $r\geq 3$, $(r-1)n\geq rk$ and let $\mathcal F_1,\ldots,\mathcal F_r\subset \genfrac{[}{]}{0pt}{}{V}{k}$, where $\genfrac{[}{]}{0pt}{}{V}{k}$ denotes the set of $k$-dimensional subspaces of $V$. Suppose that $F_1\cap\cdots\cap F_r\neq\{0\}$ holds for all $F_i\in\mathcal F_i$, $1\leq i\leq r$. Then we show that $\prod_{i=1}^r|\mathcal F_i|\leq\genfrac{[}{]}{0pt}{}{n-1}{k-1}$, provided $n-k$ is sufficiently large for fixed $q$ and $r$. Moreover, equality holds if and only if there is a common line $L$ such that every family $\mathcal F_i$ consists of all $k$-dimensional subspaces containing the line $L$. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.