##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 541.05010

**Autor: ** Erdös, Paul; Rothschild, B.; Straus, E.G.

**Title: ** Polychromatic Euclidean-Ramsey theorems. (In English)

**Source: ** J. Geom. 20, 26-35 (1983).

**Review: ** The Euclidean Ramsey Property (ERP) for a set S in Euclidean space E^{n} is that for every integer r > 0 there exists a sufficiently large integer N such that for all m \geq N and every r-coloring of E^{m} there exists a monochromatic set S' in E^{m} congruent to S. In earlier papers the first author proved that a necessary condition for ERP is that S be a finite subset of a sphere and, more generally, that if S has a k-chromatic congruent copy in all r-colorings of sufficiently high dimensional Euclidean spaces (called k-ERP), then S must be embeddable in k concentric spheres. The authors investigate sets which are exactly k-ERP (possess the k-ERP property but not (k-1)-ERP). The key to the construction of such sets is the existence of a highly transitive group of isometries (i.e., either the alternating or the symmetric group) acting on a family of subsets of a large set, and the concept of simplicial ERP introduced in the paper. The result from which essentially all other results and examples follow is: Let 0 \leq i_{1} \leq i_{2} \leq ... \leq i_{k} \leq n-1 and let P_{i} denote the set of centroids of the i-sub-simplices of a regular simplex S_{n}. Then the set S = P_{i1}\cup P_{i2}\cup...\cup P_{ik} has the exact k-ERP. An example of a set having the 3-ERP but not the 2-ERP is that consisting of the vertices of a non-obtuse, non-equilateral isosceles triangle and the trisecting points of its sides. A number of unsolved problems and conjectures are stated.

**Reviewer: ** D.Kay

**Classif.: ** * 05A17 Partitions of integres (combinatorics)

05C55 Generalized Ramsey theory

00A07 Problem books

**Keywords: ** simplicial colorings; Ramsey's theorem; Euclidean Ramsey Property

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag