Some higher-gap examples in combinatorial set theory
The authors prove the consistency of several combinatorial statements, by forcing arguments using standard cardinal collapsing. Let \(\neg SC(\kappa\) \(+)\) be the statement: there exists a sequence of functions \(<\phi_{\xi};\xi <\kappa\) \(+>\) with \(\phi_{\xi}: \xi \to | \xi |\), one-to-one, such that for all \(X\in [\kappa\) \(+]^{\omega_ 1}\) and \(\alpha <\omega_ 1\) there is \(\xi\in X\) with \(tp \phi_{\xi ''}(X\cap \xi)\geq \alpha.\) Let \(H_ n(\kappa)\) be the statement: there exists a function F: [\(\kappa\) ] \(n\to [[\kappa]^{\aleph_ 0}]^{\leq \aleph_ 0}\) such that \(\forall A\in dom(F)(\cup F(A)\subseteq \cap A)\) and \[ \forall \alpha <\omega_ 1\quad \forall X\in [\kappa]^{\omega_ 1}\quad \exists A\in [X]\quad n\quad \exists Y\subseteq X\quad (Y\in F(A)\wedge tp Y\geq \alpha). \] Amongst the consistency results obtained are the following. Theorem 1: If the existence of a Mahlo cardinal is consistent, then \(ZFC+GCH\) is consistent with \(\neg SC(\kappa\) \(+)\) for every \(\kappa\). Theorem 3: If the existence of an \(\omega\)-Mahlo cardinal is consistent, then \(ZFC+GCH\) is consistent with \(\forall n<\omega H_{n+1}(\omega_{n+1})\). Theorem 7: If the existence of an \(\omega\)-Mahlo cardinal is consistent, then \(ZFC+GCH\) is consistent with the negative partition relation \(\omega_{n+1}\nrightarrow [\omega +n+2,(\omega_ 1)_{\aleph_ 1}]^{n+2}\) \((n<\omega)\). These results are applied to prove the consistency of several old conjectures, for instance the existence of a set mapping of type \(n+2\) on \(\omega_{n+1}\) without a free set of size \(\omega_ 1\), and a conjecture on Ramsey games of \textit{A. Hajnal} and \textit{Zs. Nagy} [Trans. Am. Math. Soc. 284, 815-827 (1984; Zbl 0551.03034)].
scientific article; zbMATH DE number 4053602
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