The conjecture that a cyclic quadrilateral is not Ramsey

Let a cyclic quadrilateral be a four-point subset of the Euclidean plane whose vertices lie on a common circle, and let a finite set be Ramsey if for every positive integer $k$ there exists $d$ such that every $k$-colouring of ${\mathbb R}^d$ contains a monochromatic congruent copy of the set. For transcendental $a$ with $-1<a<1$, consider the cyclic quadrilateral with vertices

$$(-1,0),(1,0),(a,\sqrt{1-a^2}),(a,-\sqrt{1-a^2}).$$

The cyclic-quadrilateral non-Ramsey conjecture. This cyclic quadrilateral is not Ramsey. The paper proves that this quadrilateral does not embed into any finite transitive set, and earlier work showed that Ramsey sets were conjectured to be precisely the spherical sets while a rival conjecture proposed finite transitive sets as the correct characterization. The non-Ramsey conclusion itself is presented as a conjecture here.