In my previous post “transitive sets” I discussed the idea that instead of all spherical sets being Ramsey, a set is Ramsey if its symmetry group of isometries is transitive. In this post I want to give a simple example of a configuration which is spherical but can’t be embedded into a set which has a transitive symmetry group of isometries.

The example comes from the paper “Transitive sets and cyclic quadrilaterals” by I. Leader,P.A. Russell and Mark Walters. It is here. In corollary 2 of the paper they prove the set $$((-1,0),(1,0),(a,\sqrt{1-a^2},(a,-\sqrt{1-a^2}))$$ where $a$ is transcendental does not embed into any transitive set. This type of set is called a kite.

If you could find a set which satisfies the above properties and prove it is Ramsey that would provide a counterexample to the hypothesis of the first paragraph. In any case this is a good example of a set which is spherical and can’t be embedded into a transitive set. I may continue this series of posts later.