English

Sets avoiding six-term arithmetic progressions in $\mathbb{Z}_6^n$ are exponentially small

Combinatorics 2020-09-28 v1 Number Theory

Abstract

We show that sets avoiding 6-term arithmetic progressions in Z6n\mathbb{Z}_6^n have size at most 5.709n5.709^n. It is also pointed out that the "product construction" does not work in this setting, specially, we show that for the extremal sizes in small dimensions we have r6(Z6)=5r_6(\mathbb{Z}_6)=5, r6(Z62)=25r_6(\mathbb{Z}_6^2)=25 and 116r6(Z63)124 116\leq r_6(\mathbb{Z}_6^3)\leq 124.

Cite

@article{arxiv.2009.11897,
  title  = {Sets avoiding six-term arithmetic progressions in $\mathbb{Z}_6^n$ are exponentially small},
  author = {Péter Pál Pach and Richárd Palincza},
  journal= {arXiv preprint arXiv:2009.11897},
  year   = {2020}
}
R2 v1 2026-06-23T18:46:40.904Z