# 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.

File translated from T_{E}X by T_{T}H, version 1.94.

On 31 Jan 2000, 17:51.