The cap set problem and standard diagrams (Q1981684)

A cap-set is a subset of the affine space \({\mathbb Z}_3^n\) such that no three elements are on a line. In other words, a cap is a subset of \(s\) elements of \({\mathbb Z}_3^n\) such that for no three of them (say \(x,y\) and \(z\)) we have \(x+y+z=0\). In the paper under review, diagrams arising from \(s\)-caps are constructed, where by \emph{diagram} it is meant a suitable collection of points in \({\mathbb R}^2\). The use of the diagram lets the author enumerate in a systematic way the cap-sets in dimension up to \(4\) and to show that there are, up to isomorphism, \(20\) size \(18\)-caps in dimension \(4\). It is conjectured that this technique might perhaps be used to further investigate cap-sets in higher dimensions \(n\) and, perhaps, to consider the first unsolved case which is for \(n=7\).