Euclidean Ramsey theory is the study of finite configurations of points whose congruent
copies must occur monochromatically under finite colorings
of sufficiently high-dimensional Euclidean space.
A nonempty finite set 
is a Ramsey set if, for every positive integer 
, there is an 
such that every 
-coloring of 
contains a monochromatic congruent
copy of 
at its original scale (Erdős et al. 1973). The colorings need not be
measurable.
A spherical set lies on a sphere. A subtransitive set is congruent to a subset of a finite set on which its group of isometries acts transitively. Graham's spherical conjecture, recorded by Erdős et al. (1973), asks whether every finite spherical set is Ramsey.
Pálvölgyi (2026) reported a seven-point counterexample. For any transcendental number 
, the set
 |
which lies on the circle 
, is claimed to be non-Ramsey. ChatGPT supplied
the proofs and ideas. The proof uses a nonmeasurable derivation
of 
and a finite-coloring theorem. As of Oct. 8, 2026, independent review of the
claim had not been reported, and the author's disclosure explicitly excluded verification
of further claims in the appendix.
