arXiv subjects

Yongjiang Wu

Publications and source records attributed to Yongjiang Wu.

17 recordsLinked to original sources

arXiv2026

A complete solution to the Tokushige measure conjecture and its stability

We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0 (1-\varepsilon)^2(p_1p_2)^t$, then there exists $T\in\binom{[n]}{t}$ such that $μ_{p_i}(\mathcal F_i\mathbin{\triangle}\mathcal S_T)<C\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\sqrt{\varepsilon}$ estimate to $C\varepsilon$. For integer sequences, we prove that if every sequence in $\mathcal H_1\subseteq[m]^n$ agrees with every sequence in $\mathcal H_2\subseteq[m]^n$ in at least $t$ coordinates, then $|\mathcal H_1||\mathcal H_2|\leq m^{2(n-t)}$ for all $n\geq t\geq1$ and $m\geq t+1$. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection--union product theorem.

math.CO↗

arXiv2026

The product measures of cross $t$-intersecting families

We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $ t$-intersecting families, then $μ_{p_1}(\mathcal{F}_1)μ_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. Secondly, we study the intersection problems for integer sequences by proving that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with $ m > t+1$, then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. These results confirm two classical conjectures of Tokushige. As an application, we strengthen a recent theorem of Frankl--Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if $ p \geq \frac{1}{2}$ and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{μ_{p}(\mathcal{F}_1),μ_{p}(\mathcal{F}_2)\right\} \leq μ_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede--Katona.

math.CO↗

arXiv2026

Rigidity and stability for biased cross-intersecting families

Let $\mathbf p=(p_1,\ldots,p_n)$ and $\mathbf q=(q_1,\ldots,q_n)$ belong to $(0,1/2]^n$, and let $μ_{\mathbf p}$ and $μ_{\mathbf q}$ be the associated measures on $2^{[n]}$. Suppose that $p_1q_1=\max_{i\in[n]}p_iq_i$. We prove that every pair of cross-intersecting families $\mathcal A,\mathcal B\subseteq2^{[n]}$ satisfies the sharp inequality $μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\leq p_1q_1$. This confirms a conjecture of Suda, Tanaka and Tokushige [Math. Program. 166 (2017) 113--130]. We also determine all equality cases. When $p_1q_1<1/4$, equality is attained only when both families consist of all subsets containing the same product-maximizing coordinate. At the endpoint $p_1q_1=1/4$, we identify precisely the additional extremal pairs, which are induced by half-sized increasing families on the coordinates satisfying $p_i=q_i=1/2$. We further resolve the remaining conjecture from the same paper by proving a dimension-free stability theorem. Assume that the first coordinate has maximum probability under both measures and that $p_1,q_1<1/2$. If $μ_{\mathbf p}(\mathcal A)μ_{\mathbf q}(\mathcal B)\geq(1-\varepsilon)p_1q_1$, then there exists a coordinate $j$ such that both $\mathcal A$ and $\mathcal B$ are within $c(p_1,q_1)\varepsilon$, in their respective measures, of the family of all subsets containing $j$. This improves the conjectured $O(\sqrt{\varepsilon})$ bound to a linear one. The main new ingredient in the sharp measure theorem is a log-odds interpolation combined with induction on coordinate sections, while stability follows from a semidefinite estimate and a one-coordinate approximation theorem.

math.CO↗

arXiv2026

The binomial norm of intersecting-union families

In a 2021 survey on Katona's circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp binomial norm bound $ \lVert\mathcal F\rVert_n :=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1} \leq \frac{n+1}{6}. $ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof develops a continuous $p$-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for $ μ_p(\mathcal F)+μ_{1-p}(\mathcal F). $ Integrating this estimate over $p$ converts it directly into the desired binomial norm bound and recovers the optimal coefficient $\frac{1}{6}$. This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.

math.CO↗

arXiv2026

Non-uniform pairwise cross $t$-intersecting families

Let $ n\geq t\geq 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geq t$ holds for any $A_i\in \mathcal{A}_i$ and $A_j\in \mathcal{A}_j$ with $i\neq j$. In the case where $m=2$ and $\mathcal{A}_1=\mathcal{A}_2$, determining the maximum size $M(n,t)$ of a non-uniform $t$-intersecting family of sets over $[n]$ was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ are non-empty pairwise cross $t$-intersecting families, then $$ \sum_{i=1}^m |\mathcal{A}_i| \leq \max \left\{ \sum_{k=t} ^{n}\binom{n}{k} + m - 1, \, m M(n, t) \right\}. $$ Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

math.CO↗

arXiv2026

Random partition for Tokushige's $r$-wise intersecting conjecture

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $μ_{\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\mathcal A\subseteq 2^{[n]}$ is $r$-wise intersecting if $A_1\cap\cdots\cap A_r\neq\emptyset$ for all $A_1,\ldots,A_r\in\mathcal A$. In 2022, Tokushige proved that if $p_2<\frac{r-1}{r}$, then every $r$-wise intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies $μ_{\mathbf p}(\mathcal{A})\le p_1$, with equality only for stars centred at coordinates of maximum probability. He conjectured that the hypothesis $p_2<\frac{r-1}{r}$ can be replaced by $p_{r+1}<\frac{r-1}{r}$. In this paper, we prove this conjecture in full. The key novelty is the introduction of a new random partition method, which reduces the problem to at most $r$ coordinates and solves it exactly, thereby fully covering all cases with multiple supercritical coordinates.

math.CO↗

arXiv2026

Improved bound on symmetric differences of intersecting families

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

math.CO↗

arXiv2026

The Suda-Tanaka-Tokushige conjecture for $\mathbf{p}$-biased intersecting families

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge\cdots\ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal A\subseteq 2^{[n]}$ satisfies $μ_{\mathbf{p}}(\mathcal A)\le p_1$, where $μ_{\mathbf{p}}$ is the non-uniform product measure defined by $μ_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1 > p_3$ or $p_1 < \frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i \in [n]$ with $p_i = p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A} \subseteq 2^{[n]}$ satisfies $μ_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]: T\subseteq A\}$ for some $T\in \binom{[n]}{t}$ with $\prod_{i\in T} p_i=\prod_{i=1}^t p_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

math.CO↗

arXiv2026

A Survey on the Safety and Security Threats of Computer-Using Agents: JARVIS or Ultron?

Recently, AI-driven interactions with computing devices have advanced from basic prototype tools to sophisticated, LLM-based systems that emulate human-like operations in graphical user interfaces. We are now witnessing the emergence of \emph{Computer-Using Agents} (CUAs), capable of autonomously performing tasks such as navigating desktop applications, web pages, and mobile apps. However, as these agents grow in capability, they also introduce novel safety and security risks. Vulnerabilities in LLM-driven reasoning, with the added complexity of integrating multiple software components and multimodal inputs, further complicate the security landscape. In this paper, we present a systematization of knowledge on the safety and security threats of CUAs. We conduct a comprehensive literature review and distill our findings along four research objectives: \textit{\textbf{(i)}} define the CUA that suits safety analysis; \textit{\textbf{(ii)} } categorize current safety threats among CUAs; \textit{\textbf{(iii)}} propose a comprehensive taxonomy of existing defensive strategies; \textit{\textbf{(iv)}} summarize prevailing benchmarks, datasets, and evaluation metrics used to assess the safety and performance of CUAs. Building on these insights, our work provides future researchers with a structured foundation for exploring unexplored vulnerabilities and offers practitioners actionable guidance in designing and deploying secure Computer-Using Agents.

cs.CL↗

arXiv2026

Two results on set families: sturdiness and intersection

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness $β(\mathcal{F})$ is defined as the minimum $|\mathcal{F} (i,\bar{j})|$ over all $i\neq j$. A family $\mathcal{F}$ is called an IU-family if it satisfies the intersection constraint: $F\cap F'\neq \emptyset $ for all $F,F'\in \mathcal{F}$, as well as the union constraint: $F\cup F' \neq [n]$ for all $F,F'\in \mathcal{F}$. The well-known IU-Theorem states that every IU-family $\mathcal{F}\subseteq 2^{[n]}$ has size at most $ 2^{n-2}$. In this paper, we prove that if $\mathcal{F}\subseteq 2^{[n]}$ is an IU-family, then $β(\mathcal{F})\le 2^{n-4}$. This confirms a recent conjecture proposed by Frankl and Wang. As the second result, we establish a tight upper bound on the sum of sizes of cross $t$-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.

math.CO↗

arXiv2026

Subspace variations of the weighted skew Bollobás theorem

Let $V$ be a finite-dimensional real vector space. A collection $\mathcal{P} = \{(A_i,B_i)\}_{i=1}^m$ of pairs of subspaces of $V$ is called a skew Bollobás system if $\dim(A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim(A_i\cap B_j)>0$ for all $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ satisfying $ A_i = \bigoplus_{k=1}^r (A_i \cap V^{(k)})$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V^{(k)})$ for each $i\in [m]$. Denote $a_{i,k} = \dim(A_i \cap V^{(k)})$ and $b_{i,k} = \dim(B_i \cap V^{(k)})$. Suppose that $a_{1,k} \le \cdots \le a_{m,k}$ and $b_{1,k} \ge \cdots \ge b_{m,k}$ for each $k\in [r]$. Using the exterior algebraic method developed by Lovász and Scott--Wilmer, we prove that $$ \sum_{i=1}^{m} \frac{1}{\prod_{k=1}^{r} \binom{a_{i,k}+b_{i,k}}{a_{i,k}}} \le 1 . $$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Hegedüs (AJC, 2015) concerning projective subspaces, showing that any skew Bollobás system of projective subspaces in an $n$-dimensional projective space contains at most $2^{n+1} - 2$ pairs. Thirdly, we prove that if $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ This gives an extension to the subspace setting of the results of Hegedüs--Frankl (EUJC, 2024) and Yue (DM, 2026). Finally, we extend the above inequality to systems of $d$-tuples of subspaces, giving a unified bound that implies the corresponding results for $d$-tuples of subsets.

math.CO↗

arXiv2025

Metamorphic Testing for Audio Content Moderation Software

The rapid growth of audio-centric platforms and applications such as WhatsApp and Twitter has transformed the way people communicate and share audio content in modern society. However, these platforms are increasingly misused to disseminate harmful audio content, such as hate speech, deceptive advertisements, and explicit material, which can have significant negative consequences (e.g., detrimental effects on mental health). In response, researchers and practitioners have been actively developing and deploying audio content moderation tools to tackle this issue. Despite these efforts, malicious actors can bypass moderation systems by making subtle alterations to audio content, such as modifying pitch or inserting noise. Moreover, the effectiveness of modern audio moderation tools against such adversarial inputs remains insufficiently studied. To address these challenges, we propose MTAM, a Metamorphic Testing framework for Audio content Moderation software. Specifically, we conduct a pilot study on 2000 audio clips and define 14 metamorphic relations across two perturbation categories: Audio Features-Based and Heuristic perturbations. MTAM applies these metamorphic relations to toxic audio content to generate test cases that remain harmful while being more likely to evade detection. In our evaluation, we employ MTAM to test five commercial textual content moderation software and an academic model against three kinds of toxic content. The results show that MTAM achieves up to 38.6%, 18.3%, 35.1%, 16.7%, and 51.1% error finding rates (EFR) when testing commercial moderation software provided by Gladia, Assembly AI, Baidu, Nextdata, and Tencent, respectively, and it obtains up to 45.7% EFR when testing the state-of-the-art algorithms from the academy.

cs.SE↗

arXiv2025

Maximal intersecting families revisited

The well-known Erdős--Ko--Rado theorem states that for $n> 2k$, every intersecting family of $k$-sets of $[n]:=\{1,\ldots ,n\}$ has at most $ {n-1 \choose k-1}$ sets, and the extremal family consists of all $k$-sets containing a fixed element (called a full star). The Hilton--Milner theorem provides a stability result by determining the maximum size of a uniform intersecting family that is not a subfamily of a full star. The further stabilities were studied by Han and Kohayakawa (2017) and Huang and Peng (2024). Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. Let $k \geq 1, t\ge 0$ and $n \geq 2 k+t$ be integers. Frankl (2016) proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Recently, Wu (2023) sharpened Frankl's result by establishing a stability variant. The aim of this paper is two-fold. Inspired by the above results, we first prove a further stability variant that generalizes both Frankl's result and Wu's result. Secondly, as an interesting application, we illustrate that the aforementioned results on cross-intersecting families could be used to establish the stability results of the Erdős--Ko--Rado theorem. More precisely, we present new short proofs of the Hilton--Milner theorem, the Han--Kohayakawa theorem and the Huang--Peng theorem. Our arguments are more straightforward, and it may be of independent interest.

math.CO↗

arXiv2025

Stabilities of the Kleitman diameter theorem

Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erdős, Kleitman determined the maximum size of a family with fixed diameter, which states that a family with diameter $s$ has cardinality at most that of a Hamming ball of radius $s/2$. Specifically, if $\mathcal{F} \subseteq 2^{[n]}$ is a family with diameter $s$, then for $s=2d$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i}$; for $s=2d+1$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i} + {n-1 \choose d}$. This result is known as the Kleitman diameter theorem, which generalizes both the Katona union theorem and the Erdős--Ko--Rado theorem. In 2017, Frankl provided a complete characterization of the extremal families of Kleitman's theorem and provided a stability result. In this paper, we determine the extremal families of Frankl's theorem and establish a further stability result of Kleitman's theorem. This solves a recent problem proposed by Li and Wu. Our findings constitute the second stability for the Kleitman diameter theorem.

math.CO↗

arXiv2025

A result for hemi-bundled cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for $k \geq 1, t\ge 0$ and $n \geq 2 k+t$, if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, in which $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. This bound can be attained when $\mathcal{F}$ consists of a single set. In this paper, we generalize this result under the constraint $|\mathcal{F}| \geq r$ for every $r\leq n-k-t+1$. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the $s$-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.

math.CO↗

arXiv2025

Proof of Frankl's conjecture on cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $\{1,2,\ldots,n\}$. Let $t, s, k, n$ be non-negative integers with $k \geq s+1$ and $n \geq 2 k+t$. In 2016, Frankl proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $|\mathcal{F}| \geq 1$, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Furthermore, Frankl conjectured that under an additional condition $\binom{[k+t+s]} {k+t}\subseteq\mathcal{F}$, the following inequality holds: $$ |\mathcal{F}|+|\mathcal{G}| \leq\binom{k+t+s}{k+t}+\binom{n}{k}-\sum_{i=0}^s\binom{k+t+s}{i}\binom{n-k-t-s}{k-i}. $$ In this paper, we prove this conjecture. The key ingredient is to establish a theorem for cross-intersecting families with a restricted universe. Moreover, we derive an analogous result for this conjecture.

math.CO↗

arXiv2024

Snevily's Conjecture about $\mathcal{L}$-intersecting Families on Set Systems and its Analogue on Vector Spaces

The classical Erdős-Ko-Rado theorem on the size of an intersecting family of $k$-subsets of the set $[n] = \{1, 2, \dots, n\}$ is one of the fundamental intersection theorems for set systems. After the establishment of the EKR theorem, many intersection theorems on set systems have appeared in the literature, such as the well-known Frankl-Wilson theorem, Alon-Babai-Suzuki theorem, and Grolmusz-Sudakov theorem. In 1995, Snevily proposed the conjecture that the upper bound for the size of an $\mathcal{L}$-intersecting family of subsets of $[n]$ is ${{n} \choose {s}}$ under the condition $\max \{l_{i}\} < \min \{k_{j}\}$, where $\mathcal{L} = \{l_{1}, \dots, l_{s}\}$ with $0 \leq l_{1} < \cdots < l_{s}$ and $k_{j}$ are subset sizes in the family. In this paper, we prove that Snevily's conjecture holds for $n \geq {k^{2} \choose {l_{1}+1}}s + l_{1}$, where $k$ is the maximum subset size in the family. We then derive an analogous result for $\mathcal{L}$-intersecting families of subspaces of an $n$-dimensional vector space over a finite field $\mathbb{F}_{q}$.

math.CO↗