A Bollob\'as-type theorem for affine subspaces

TL;DR

This paper establishes an upper bound similar to Bollobás' theorem for families of affine subspaces in finite fields, providing near-optimal constructions to demonstrate the bound's tightness.

Contribution

It introduces a Bollobás-type upper bound for affine subspaces and constructs near-sharp examples, extending combinatorial bounds to affine geometry over finite fields.

Findings

Proved a Bollobás-type upper bound for affine subspaces.

Constructed families of affine subspaces close to the bound.

Showed the bound is nearly optimal with explicit examples.

Abstract

Let $W$ denote the $n$-dimensional affine space over the finite field $\mathbb F_q$. We prove here a Bollob\'as-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.