Finding large 3-free sets I: The small n case

摘要

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, present techniques to find large 3-free sets of {1,…,n} for n⩽250, and give empirical results obtained by coding up those techniques. In the sequel we survey the known techniques for finding large 3-free sets of {1,…,n} for large n, discuss variants of them, and give empirical results obtained by coding up those techniques and variants.