Transitive Sets and Cyclic Quadrilaterals

TL;DR

This paper demonstrates that most cyclic quadrilaterals cannot be embedded into transitive sets, providing the first explicit examples of spherical sets with this property, motivated by Euclidean Ramsey theory questions.

Contribution

It proves that almost all cyclic quadrilaterals do not embed into any transitive set and provides explicit examples, advancing understanding in geometric and Ramsey theory.

Findings

Most cyclic quadrilaterals cannot embed into transitive sets.

Explicit examples of spherical sets not embeddable into transitive sets are provided.

This work addresses questions in Euclidean Ramsey theory.

Abstract

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.