A theorem and some consistency results in partition calculus
For ordinals \(\alpha\), \(\beta\), \(\gamma\) (the order type is concerned, not the cardinality) say \(\alpha \to (\beta,\gamma)^ 2\) iff whenever \(f:[\alpha]^ 2\to 2\) there is an \(x\subset \alpha\) s.t. either x has order type \(\beta\) and \(\forall (a,b)\in [x]^ 2\) \(f(a,b)=0\) or x has order type \(\gamma\) and \(\forall (a,b)\in [x]^ 2\) \(f(a,b)=1.\) It is shown that for regular \(\kappa\) such that \(\forall \lambda <\kappa\), \(\lambda^{\aleph_ 0}<\kappa\) and \(k<\omega\kappa \omega \to (\kappa \omega,k)^ 2\). Various consistency results are given, e.g.: \(ZFC+\omega_ 2\omega \to (\omega_ 2\omega,3)^ 2+\omega_ 3\omega_ 1\nrightarrow (\omega_ 3\omega_ 1,3)^ 2\). These are extensions of the Erdős-Rado and Erdős-Dushnik-Miller theorems.
scientific article; zbMATH DE number 4014699
scientific article