### Identities on cardinals less than $\aleph_\omega$

by Gilchrist and Shelah. [GcSh:491]

J Symbolic Logic, 1996

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 Erdos-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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