The Leader-Russell-Walters conjecture: is every Euclidean Ramsey set subtransitive?

The manuscript proves that every nonempty set of at most five points on a circle is Ramsey, so every cyclic quadrilateral is Ramsey; in particular the kites $K_a$ with $a\in(-1,1)$ transcendental are Ramsey. With Leader, Russell and Walters' theorem that these kites are not subtransitive, this disproves the necessity direction of Conjecture A and disproves Conjecture 3. The sufficiency direction (subtransitive implies Ramsey) is proved in the same paper, so the conjecture fails in one direction only. The argument depends on the classification theorem of the same manuscript and on the cited non-subtransitivity result, which is not reproved.

The manuscript proves that every nonempty set of at most five points on a circle is Ramsey, so every cyclic quadrilateral is Ramsey; in particular the kites $K_a$ with $a\in(-1,1)$ transcendental are Ramsey. With Leader, Russell and Walters' theorem that these kites are not subtransitive, this disproves the necessity direction of Conjecture A and disproves Conjecture 3. The sufficiency direction (subtransitive implies Ramsey) is proved in the same paper, so the conjecture fails in one direction only. The argument depends on the classification theorem of the same manuscript and on the cited non-subtransitivity result, which is not reproved.