Treffer: Asymptotic Behavior of r sub (k(N ))(N ): Progression-Free Sets with Growing Length

This paper determines the asymptotic behavior of the function r_k(N), which represents the maximum size of a subset of the integers from 1 to N that contains no nontrivial arithmetic progression of length k. Unlike previous results that focused on fixed values of k, we analyze the case where k grows slowly with N. Using a combinatorial framework based on clause saturation and density arguments, we prove that if k exceeds a constant times log N divided by log log N, then any subset of positive density must contain a k-term arithmetic progression. This resolves a major open question originally posed by Erdős and extends classical results by Roth and Szemerédi. The argument avoids analytical or Fourier-based methods and is entirely elementary and combinatorial in nature.