Transitive Sets and Cyclic Quadrilaterals

Motivated by some questions in Euclidean Ramsey theory, our aim in this note is to show that there exists a cyclic quadrilateral that does not embed into any transitive set (in any dimension). We show that in fact this holds for almost all cyclic quadrilaterals, and we also give explicit examples of such cyclic quadrilaterals. These are the first explicit examples of spherical sets that do not embed into transitive sets.