Cross $t$-intersecting families for finite affine spaces

Abstract

Denote the collection of all $k$-flats in $AG(n,{\mathbb{F}}_q)$ by $\mathcal{M}(k,n)$. Let ${\mathcal{F}}_1\subset \mathcal{M}(k_1,n)$ and ${\mathcal{F}}_2\subset \mathcal{M}(k_2,n)$ satisfy $\dim (F_1\cap F_2)\ge t$ for any $F_1\in {\mathcal{F}}_1$ and $F_2\in {\mathcal{F}}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-flats in $AG(n,{\mathbb{F}}_q)$. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.

Some typos are corrected