Cross $t$-intersecting families for finite affine spaces

TL;DR

This paper investigates the structure of cross $t$-intersecting families of flats in finite affine spaces, proving that those with maximum product of sizes are trivial and characterizing non-trivial extremal families.

Contribution

It establishes that maximum product cross $t$-intersecting families are trivial and characterizes the structure of non-trivial extremal families in finite affine spaces.

Findings

Maximum product cross $t$-intersecting families are trivial.

Characterization of non-trivial extremal families.

Structural results in finite affine space combinatorics.

Abstract

Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by $\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and $\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. 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.