---

Source. Published pp. 6–7, Problem 6.2, Corollary 6.5 and Open Problem 6.6. The questions below are historical statements, not newly verified open research targets.

Problem 6.2 asks whether there are Ramsey sets which are not sphere Ramsey. Open Problem 6.6 asks whether obtuse triangles are hyper-Ramsey. The paper also notes that its methods do not then prove the hyper-Ramsey property for all nondegenerate simplices. These distinctions use definitions; ordinary Ramsey, super-Ramsey and hyper-Ramsey are different claims.

Later published evidence. Frankl and Rödl, Strong Ramsey properties of simplices, Israel Journal of Mathematics 139 (2004), 215–236, states Theorem 3.3, “Every simplex is hyper Ramsey,” on printed p. 221 (published PDF). Its Definition 3.1 uses witnesses at radius $\sqrt{\rho(A)^2+\alpha}$ for every $\alpha>0$; this is the same family of positive radius slacks as $\rho(A)+\delta$ under the substitution $\alpha=2\rho(A)\delta+\delta^2$. Thus the published later theorem addresses the obtuse-triangle question and the more general simplex limitation. This page records the exact external statement; the 2004 proof has not been independently reconstructed here. Do not carry the 1990 label “Open Problem” forward as its present status.

The 2004 paper's Theorem 3.2 states the hyper-Ramsey property for boxes. That statement alone does not fill the arbitrary-subset radius step identified in corollary_6_5. A separate proof or exact later theorem is needed for that general clause.

Current scope. The broader Ramsey/sphere-Ramsey comparison and the classification of spherical configurations require their own dated synthesis, including recent work such as Moore's pyramid closure and Mirabi's distinct extension proof. Neither this historical question inventory nor the failure to locate a proof in these particular sources establishes present openness. No problem-page status is changed by this source compilation.

Bears on. #174.