Simplices in $t$-intersecting families for vector spaces
Haixiang zhang, Mengyu Cao, Mei Lu, Jiaying Song
TL;DR
This work determines the maximal number of $(r+1,t)$-simplices in a $k$-uniform $r$-wise $t$-intersecting family of $k$-subspaces of an $n$-dimensional vector space over a finite field. The authors introduce the $t$-covering number and a projection framework to analyze the extremal structure, proving that the maximum is $n_{t+r,k}$ and that equality occurs precisely when the family lies between the subspace families $\mathcal{F}^*_{X,k}$ and $\mathcal{F}_{X,k}$ for some $(t+r)$-subspace $X$, provided $n \ge 3kr^2+3krt$ and $k \ge r+t-1$. They also deduce corollaries for the $n$-threshold case of triangles ($r=2$, $t=1$) with $n \ge 2k+9$. The results extend vector-space analogues of classical extremal set theory, offering a tight structural characterization via the $t$-covering approach and Gaussian-binomial combinatorics, with implications for triangle-like configurations in subspace families.
Abstract
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 $V$. A family $\mathcal{F}\subseteq {V\brack k}$ is called $k$-uniform $r$-wise $t$-intersecting if for any $F_1, F_2, \dots, F_r \in \mathcal{F}$, we have $\dim\left(\bigcap_{i=1}^r F_i \right) \geq t$. An $r$-wise $t$-intersecting family $\{X_1, X_2, \dots, X_{r+1}\}$ is called a $(r+1,t)$-simplex if $\dim\left(\bigcap_{i=1}^{r+1} X_i \right) < t$, denoted by $Δ_{r+1,t}$. Notice that it is usually called triangle when $r=2$ and $t=1$. For $k \geq t \geq 1$, $r \geq 2$ and $n \geq 3kr^2 + 3krt$, we prove that the maximal number of $Δ_{r+1,t}$ in a $k$-uniform $r$-wise $t$-intersecting subspace family of $V$ is at most $n_{t+r,k}$, and we describe all the extreme families. Furthermore, we have the extremal structure of $k$-uniform intersecting families maximizing the number of triangles for $n\geq 2k+9$ as a corollary.