## 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 En 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 Em there exists a monochromatic set S' in Em 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 i1 \leq i2 \leq ... \leq ik \leq n-1 and let Pi denote the set of centroids of the i-sub-simplices of a regular simplex Sn. Then the set S = Pi1\cup Pi2\cup...\cup Pik 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

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