# Character-free approach to progression-free sets

## TL;DR

This paper introduces a simple combinatorial method to estimate the maximum density of sets without arithmetic progressions in finite vector spaces, showing it scales inversely with the dimension.

## Contribution

It provides an elementary combinatorial proof that the density of progression-free sets in finite vector spaces is bounded by a constant times 1/r, offering a new perspective.

## Key findings

- Progression-free set density is O(1/r) in finite vector spaces.
- Elementary combinatorial argument suffices for the bound.
- Simplifies previous complex proofs.

## Abstract

We present an elementary combinatorial argument showing that the density of a progression-free set in a finite r-dimensional vector space is O(1/r).