Suggestion of a problem

Karl Svozil
Institut für Theoretische Physik, Technische Universität Wien
Wiedner Hauptstraß e 8-10/136, A-1040 Vienna, Austria
e-mail: svozil@tuwien.ac.at

http://tph.tuwien.ac.at/[    \tilde] svozil/publ/probsug.{htm,ps,tex}

Abstract

I propose a coloring problem which, to my knowledge, has not been solved yet.

(a) Find the chromatic number of the unit sphere in n > 3 dimensional real Hilbert space.

(b) Find the chromatic number of the rational unit sphere in n > 3 dimensions.

(c) Find the minimum number of vectors in three dimensional real Hilbert space which orthogenerates a system of vectors whose chromatic number exceeds three. Generalize this result to n Dimensions.

(d) Find the minimum number of vectors in three dimensional real Hilbert space which orthogenerates a system of vectors for which no valuation (two-valued measure) exists such that every orthogonal tripod contains exactly two vectors whose measure is 0 and one vector whose measure is 1. Generalize this result to n Dimensions.

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On 31 Jan 2000, 17:51.