A GCH Example of an Ordinal Graph with no Infinite Path

Abstract

It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis. Erdös, Hajnal and Milner have proved that limit ordinals $\alpha < \omega _1^{\omega + 2}$ satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes $\alpha \to {(\alpha , \operatorname {infinite} \operatorname {path} )^2}$ to mean that every graph on an ordinal $\alpha$ either has a subset order isomorphic to $\alpha$ in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality ${\aleph _m}$ for $m$ a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality ${\aleph _m}$ for which the set mapping theorem is true. In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality ${\aleph _m}$, where $m$ is a natural number at least two.