Finding Large Sets Without Arithmetic Progressions of Length Three: An Empirical View and Survey II

TL;DR

This paper reviews the construction of large 3-free sets, compares empirical results with theoretical methods, and investigates how small n sets relate to asymptotic bounds, providing practical insights into their sizes.

Contribution

It offers an empirical analysis of large 3-free sets for small n and compares different construction methods to assess their effectiveness.

Findings

Empirical sizes of 3-free sets for small n are presented.

Comparison of literature methods shows when asymptotic advantages manifest.

Insights into the practical construction of large 3-free sets are provided.

Abstract

There has been much work on the following question: given n how large can a subset of {1,...,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch applications of large 3-free sets, review the literature of how to construct large 3-free sets, and present empirical studies on how large such sets actually are. The two main questions considered are (1) How large can a 3-free set be when n is small, and (2) How do the methods in the literature compare to each other? In particular, when do the ones that are asymptotically better actually yield larger sets? (This paper overlaps with our previous paper with the title { Finding Large 3-Free Sets I: the Small n Case}.)