##
**Zentralblatt MATH**

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

**Zbl.No: ** 419.04002

**Autor: ** Baumgartner, James E.; Erdös, Paul; Galvin, Fred; Larson, J.

**Title: ** Colorful partitions of cardinal numbers. (In English)

**Source: ** Can. J. Math. 31, 524-541 (1979).

**Review: ** Let \kappa, \lambda, \mu, \nu be infinite cardinal numbers. Let [\kappa]^{2} denote the set of all two element subsets of \kappa, and consider [\kappa]^{2} as the set of edges for the complete graph on \kappa vertices. The authors define the relation CP(\kappa,\mu,\nu) to hold if there is an edge colouring R: [\kappa]^{2} ––> \mu with \mu colours such that for every proper \nu size subset X of \kappa there is a vertex x in \kappa-X such that the edges between x and the vertices in X receive at least **max**(\mu\nu) colours. The relation CP^{\sharp}(\kappa,\mu,\nu) holds if there is such a colouring which is one-to-one on the edges between x and the vertices in X. There are related properties BP and BP^{\sharp}, where BP(\kappa,\lambda,\mu,\nu) holds if there is a colouring R: \kappa×\lambda ––> \mu of the complete \kappa, \lambda bipartite graph with \mu colours, such that for every \nu size subset X of \kappa there is a point x in \lambda such that the edges between x and the vertices in X receive at least **max**(\mu,\nu) colours. The paper is devoted to a discussion of the properties BP and BP^{\sharp}. From these, properties of CP and CP^{\sharp} are deduced, sufficient to characterize completely CP and CP^{\sharp} under the assumption of the generalized continuum hypothesis.

**Reviewer: ** N.H.Williams

**Classif.: ** * 04A20 Combinatorial set theory

04A10 Ordinal and cardinal numbers; generalizations

04A30 Continuum hypothesis and generalizations

05C15 Chromatic theory of graphs and maps

**Keywords: ** infinite graphs; edge colourings; infinite cardinal numbers; generalized continuum hypothesis

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag