A Constructive Lower Bound on Szemer\'edi's Theorem

TL;DR

This paper introduces a new constructive method to establish lower bounds on the size of large sets avoiding k-term arithmetic progressions, improving known bounds for large n and linking to Erdős's conjecture.

Contribution

It provides the first constructive lower bound on r_k(n) that approaches the upper bounds as k increases, advancing understanding of progression-free sets.

Findings

Established a lower bound of n^{1 - c_k/(k ln k)} for r_k(n) with c_k → 1 as k → ∞

Demonstrated the potential to resolve Erdős's conjecture with tighter bounds on r_k(n)

Improved bounds are valid over increasingly large ranges of n for larger k

Abstract

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound $n^{1-\frac{c_k}{k \ln k}} < r_k(n)$ where $k$ is prime, and $c_k \rightarrow 1$ as $k \rightarrow \infty$. This bound is the best known for an increasingly large interval of $n$ as we choose larger and larger $k$. We also demonstrate that one can prove or disprove a conjecture of Erd\H{o}s on arithmetic progressions in large sets once tight enough bounds on $r_k(n)$ are obtained.