Finding Large Sets Without Arithmetic Progressions of Length Three: An Empirical View and Survey II
William Gasarch, James Glenn, Clyde Kruskal
TL;DR
The paper surveys the problem of finding large 3-free subsets of {1,...,n}, balancing empirical findings for small n with survey and implementation of asymptotically optimal constructions. It documents several constructive and nonconstructive methods (Base 3, Base 5, KD, Block, and Sphere methods), with the Sphere family emerging as the leading asymptotic approach, albeit nonconstructive. Through extensive computational work, it reports exact sz(n) up to n=186 and tight bounds up to n=250, and it analyzes how these methods compare across n, showing that asymptotically best results often outpace smaller-scale constructions but that for practical finite n other methods can compete. The work connects Szemerédi-type limits to applications in combinatorics and CS, demonstrates practical coding of sphere-based constructions, and outlines future directions to tighten bounds and extend to higher-length progressions. Overall, it provides a comprehensive empirical and methodological map of constructing and bounding large 3-free sets with implications for related algorithmic and complexity problems.
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}.)