Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups

Abstract

A subset S of an abelian group G is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, S is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.