Equilateral dimension of the rectilinear space (Q5926319)

The equilateral dimension \(e(M)\) of a metric space \(M\) is the maximum cardinality of an equilateral set (any two distinct points in that set are at the same distance). It has been conjectured that \(e(\ell_1(k))=2k\) for all \(k\geq 1\), where \(\ell_1(k)\) stands for \({\mathbb R}^k\) equipped with the \(\ell_1\)-norm (Manhattan metric). This conjecture has been proved for \(k\leq 3\), and the authors prove it for \(k=4\) and study several related problems.