A diamond example of an ordinal graph with no infinite paths
Given ordinals \(\alpha\) and \(\beta\), \(\alpha\to (\beta\), infinite \(path)^ 2\) means that every graph on \(\alpha\) either contains an independent (edge-free) set of order type \(\beta\) or an infinite path. It is easy to see that \(\alpha +n+1\nrightarrow (\alpha +1\), infinite \(path)^ 2\) whenever \(\alpha >0\) and \(n<\omega\). \textit{P. Erdős}, \textit{A. Hajnal} and \textit{E. C. Milner} [Combinat. Theory Appl., Colloq. Math. Soc. János Bolyai 4, 327-363 (1970; Zbl 0215.329)] have shown that \(\alpha\to (\alpha\), infinite \(path)^ 2\) for all limit ordinals \(\alpha <\omega_ 1^{\omega +2}\). This positive result has been extended by the second author [in the paper reviewed below (Zbl 0703.03029)] to all limit ordinals \(\alpha <2^{\aleph_ 0}\) under the assumption of Martin's Axiom. The authors show that if Jensen's principle \(\diamondsuit\) holds, then \(\alpha \nrightarrow (\omega_ 1^{\omega +2}\), infinite \(path)^ 2\) for all \(\alpha <\omega_ 2\).
scientific article; zbMATH DE number 4152364