Simplices in $t$-intersecting families for vector spaces

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$ and $\left[\genfrac{}{}{0ex}{}{V}{k}\right]$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[\genfrac{}{}{0ex}{}{V}{k}\right]$ is called $k$-uniform $r$-wise $t$-intersecting if for any $F_1,F_2,\dots ,F_r\in \mathcal{F}$, we have $\dim \left({\bigcap }_{i=1}^r{F}_i\right)\ge t$. 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 $\hat{I}{”}_{r+1,t}$. Notice that it is usually called triangle when $r=2$ and $t=1$. For $k\ge t\ge 1$, $r\ge 2$ and $n\ge 3k{r}^2+3krt$, we prove that the maximal number of $\hat{I}{”}_{r+1,t}$ 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 $n\ge 2k+9$ as a corollary.

18 pages