Identities on cardinals less than aleph_omega

Details Identities on cardinals less than aleph_omega Identities on cardinals less than aleph_omega

1995-05-15

arxiv.org/abs/math/9505215v1

J. Symbolic Logic 61 (1996), 780--787

Mathematics

Logic

Scientific paper

Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erd\H{o}s-Rado theorem implies that there is an infinite monochromatic subgraph. However, if kappa <= 2^{aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when kappa <= 2^{aleph_0}. In particular, we are concerned with the case aleph_1 <= kappa <= aleph_omega

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