Euclidean Ramsey theorems on the n-sphere

✍ Scribed by H. L. Graham

• Publisher: John Wiley and Sons
• Year: 1983
• Tongue: English
• Weight: 340 KB
• Volume: 7
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.3190070114

✦ Synopsis

Abstract

Let us call a finite subset X of a Euclidean m‐space E^m^ Ramsey if for any positive integer r there is an integer n = n(X;r) such that in any partition of E^n^ into r classes C~1~,…, C~r~, some C~i~ contains a set X' which is the image of X under some Euclidean motion in E^n^. Numerous results dealing with Ramsey sets have been proved in recent years although the basic problem of characterizing the Ramsey sets remains unsettled. The strongest constraints currently known are: (i) Any Ramsey set must lie on the surface of some sphere; (ii) Any subset of the set of vertices of a rectangular parallelepiped is Ramsey. In this paper we examine the corresponding problem in the case that our underlying spaces are (unit) n‐spheres S^n^ and the allowed motions are orthogonal transformations of S^n^ onto itself. In particular, we show that for subsets of S^n^ which are not too “large,” results similar to (i) and (ii) hold.