Two negative partition relations
Hajnal has shown that the continuum hypothesis implies \(\omega_ 1\cdot \omega \nrightarrow (\omega_ 1\cdot \omega,3)^ 2\) and \(\omega_ 1\cdot \omega_ 1\nrightarrow (\omega_ 1\cdot \omega_ 1,3)^ 2\). Call the following set-theoretic hypotheses \(\Gamma\) and \(\Delta\) respectively: (a) There exists a family \({\mathcal A}\subseteq [\omega_ 1]^{\omega}\) such that card(\({\mathcal A})=\omega_ 1\) and \(\forall X\in [\omega_ 1]^{\omega_ 1}:\exists A\in {\mathcal A}:\) \(A\subseteq X\); (b) There exists a family \({\mathcal G}\subseteq^{\omega}\omega\) such that card(\({\mathcal G})=\omega_ 1\) and \(\forall f\in^{\omega}\omega:\exists g\in {\mathcal G}:\forall n<\omega:\) \(f(n)<g(n)\). The conjunction of \(\Gamma\) and \(\Delta\) is known to be weaker than the continuum hypothesis. In this paper it is shown that \(\omega_ 1\cdot \omega \nrightarrow (\omega_ 1\cdot \omega,3)^ 2\) follows from the conjunction of \(\Gamma\) and \(\Delta\), and \(\omega_ 1\cdot \omega_ 1\nrightarrow (\omega_ 1\cdot \omega_ 1,3)^ 2\) from \(\Gamma\).
scientific article; zbMATH DE number 4041905