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Kaishun Wang

Publications and source records attributed to Kaishun Wang.

At least 19 recordsLinked to original sources
arXiv2026

On Erdős--Ko--Rado and Hilton--Milner Theorems for Direct Products

We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erdős--Ko--Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{155} (2018), 493--502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{156} (2018), 44--60] asking for a classification of all extremal families, including the possible non-shifted maximizers.

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arXiv2026

Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}\subseteq {V\brack k}$ and $\mathcal{G}\subseteq {V\brack \ell}$ are said to be cross $t$-intersecting if $\dim(F\cap G)\geq t$ for all $F\in\mathcal{F}$ and $G\in \mathcal{G}$. In this paper, we determine the extremal structures when $|\mathcal{F}||\mathcal{G}|$ attains the maximum value under the conditions $\dim\left(\cap_{F\in \mathcal{F}}F\right)<t$ and $\dim\left(\cap_{G\in \mathcal{G}}G\right)<t$.

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arXiv2026

Extremal $t$-intersecting families for finite sets with $t$-covering number at least $t+2$

Let $\mathcal{F}\subseteq{[n]\choose k}$ be a $t$-intersecting family. Define the $t$-covering number $τ_t(\mathcal{F})$ of $\mathcal{F}$ as the minimum size of a subset $S$ of $[n]$ with $|S\cap F|\geqslant t$ for each $F\in\mathcal{F}$. In this paper, we characterize $\mathcal{F}$ for which $|\mathcal{F}|$ takes the maximum value under the condition that $τ_t(\mathcal{F})\geqslant t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.

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arXiv2026

Non-trivial cross-$t$-intersecting families for vector spaces with the maximum sum of sizes

Let $V$ be an $n$-dimensional vector space over a finite field. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are non-empty families of $k$-subspaces and $\ell$-subspaces of $V$, respectively. They are said to be cross-$t$-intersecting if $\dim(F\cap G)\geq t$ for any $F\in\mathcal{F}$ and $G\in \mathcal{G}$, and are further called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$ and $\dim(\cap_{G\in\mathcal{G}}G)<t$. In this paper, we characterize the non-trivial cross-$t$-intersecting families with the maximum sum of sizes. When $t=1$, our result serves as the $q$-analog of the theorems in [9,11].

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arXiv2026

On extremal cross $t$-intersecting families with $t$-covering number conditions

Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots,n\}$. The $t$-covering number $τ_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that $|F\cap T|\geq t$ for all $F\in \mathcal{F}$. For positive integers $k_1$ and $k_2$, let $\mathcal{F}_i$ be a collection of $k_i$-subsets of $[n]$ for $i\in \{1,2\}$. The families $\mathcal{F}_1$ and $\mathcal{F}_2$ are said to be cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for all $F_1\in\mathcal{F}_1$ and $F_2\in \mathcal{F}_2$. When $\mathcal{F}_1=\mathcal{F}_2$, $\mathcal{F}_1$ is called a $t$-intersecting family. In this paper, we first characterize the extremal structures of cross $t$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ that maximize $|\mathcal{F}_1||\mathcal{F}_2|$ under the condition that $τ_t(\mathcal{F}_1)\geq t+1$ and $τ_t(\mathcal{F}_2)\geq t+1$. We then describe the maximal $t$-intersecting families with $t$-covering number $t+1$.

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arXiv2026

$s$-almost $t$-intersecting families for finite sets

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\left|\bigcap_{F\in\mathcal{F}} F\right|<t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.

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arXiv2025

$s$-almost cross-$t$-intersecting families for finite sets

Two families $\mathcal{F}$ and $\mathcal{G}$ of $k$-subsets of an $n$-set are called $s$-almost cross-$t$-intersecting if each member in $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members in $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we characterize the $s$-almost cross-$t$-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the $s$-almost cross-$t$-intersecting families which are not cross-$t$-intersecting.

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arXiv2025

Semicomplete multipartite weakly distance-regular digraphs

A digraph is semicomplete multipartite if its underlying graph is a complete multipartite graph. As a special case of semicomplete multipartite digraphs, Jørgensen et al. \cite{JG14} initiated the study of doubly regular team tournaments. As a natural extension, we introduce doubly regular team semicomplete multipartite digraphs and show that such digraphs fall into three types. Furthermore, we give a characterization of all semicomplete multipartite commutative weakly distance-regular digraphs.

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arXiv2024

$s$-almost $t$-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in \mathcal{F}$. We say $\mathcal{F}$ is $s$-almost $t$-intersecting if for each $F\in \mathcal{F}$ there are at most $s$ members $F^{\prime}$ of $\mathcal{F}$ such that $\dim(F\cap F^{\prime})<t$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size for $(s,t)\neq(1,1)$. The result for $1$-almost $1$-intersecting families provided by Shan and Zhou is generalized.

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arXiv2024

Locally semicomplete weakly distance-regular digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood (resp. in-neighbourhood) of any vertex induces a semicomplete digraph. In this paper, we characterize all locally semicomplete weakly distance-regular digraphs under the assumption of commutativity.

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arXiv2024

A Bollobáss-type theorem on singular linear spaces

Bollobás-type theorem determines the maximum cardinality of a Bollobás system of sets. The original result has been extended to various mathematical structures beyond sets, including vector spaces and affine spaces. This paper generalizes the Bollobás-type theorem to singular linear spaces, and determine the maximum cardinality of (skew) Bollobás systems on them.

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arXiv2024

Weakly distance-regular digraphs whose underlying graphs are distance-regular,II

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [16], we classified all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes, or Doob graphs. In this paper, we classify all commutative weakly distance-regular digraphs whose underlying graphs are Johnson graphs or folded Johnson graphs.

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arXiv2024

More on $r$-cross $t$-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the extremal structure when $\prod_{i=1}^r|\mathscr{F}_i|$ is maximum under the condition that $\dim(\bigcap_{F\in\mathscr{F}_i}F)<t$ for each $i$.

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arXiv2024

Weakly distance-regular digraphs whose underlying graphs are distance-regular, I

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [8], the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project, and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes and Doob graphs, respectively.

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arXiv2023

Weakly distance-regular circulants, I

We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.

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arXiv2022

Some intersection theorems for finite sets

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap\cdots\cap F_r|\geq t$ for all $F_i\in\mathcal{F}_i\ (1\leq i\leq r),$ and said to be non-trivial if $|\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F|<t$. If the $r$-cross $t$-intersecting families $\mathcal{F}_1,\ldots,\mathcal{F}_r$ satisfy $\mathcal{F}_1=\cdots=\mathcal{F}_r=\mathcal{F}$, then $\mathcal{F}$ is well known as $r$-wise $t$-intersecting family. In this paper, we describe the structure of non-trivial $r$-wise $t$-intersecting families with maximum size, and give a stability result for these families. We also determine the structure of non-trivial $2$-cross $t$-intersecting families with maximum product of their sizes.

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