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