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

A subset $S$ of an abelian group $G$ is called $3$-$\mathrm{A}\mathrm{P}$ free if it does not contain a three term arithmetic progression. Moreover, $S$ is called complete $3$-$\mathrm{A}\mathrm{P}$ 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 $3$-$\mathrm{A}\mathrm{P}$ free set, which is necessarily complete. In this paper we are interested in the minimum size of complete $3$-$\mathrm{A}\mathrm{P}$ free sets. We define and study saturation w.r.t. $3$-$\mathrm{A}\mathrm{P}$s and present constructions of small complete $3$-$\mathrm{A}\mathrm{P}$ free sets and $3$-$\mathrm{A}\mathrm{P}$ saturating sets for several families of vector spaces and cyclic groups.