A Bollobás-type theorem for affine subspaces

Abstract

Let $W$ denote the $n$-dimensional affine space over the finite field ${\mathbb{F}}_q$. We prove here a Bollobás-type upper bound in the case of the set of affine subspaces. We give a construction of a pair of families of affine subspaces, which shows that our result is almost sharp.

7 pages