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

TL;DR

This paper proves that sets avoiding six-term arithmetic progressions in the group ^n are exponentially small, with size at most 5.709^n, and discusses limitations of product constructions in this setting.

Contribution

It establishes an exponential upper bound on the size of progression-avoiding sets in ^n and analyzes the extremal sizes in small dimensions, highlighting the failure of product constructions.

Findings

Sets avoiding 6-term progressions have size at most 5.709^n.

Extremal sizes in small dimensions are r_6()=5, r_6(^2)=25.

Product construction does not produce extremal sets in this setting.

Abstract

We show that sets avoiding 6-term arithmetic progressions in $\mathbb{Z}_6^n$ have size at most $5.709^n$. 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 $r_6(\mathbb{Z}_6)=5$, $r_6(\mathbb{Z}_6^2)=25$ and $ 116\leq r_6(\mathbb{Z}_6^3)\leq 124$.