Was Sierpinski right? I
Motivated by the result of \textit{S. Todorcevic} [Acta. Math. 159, 261-294 (1987)] that \(\aleph_ 1\nrightarrow [\aleph_ 1]^ 2_{\aleph_ 1}\), the author provides a few results that complement it. We mention the principal results: 1. Assume that V satisfies GCH and that \(\mu <\kappa <\lambda\) are regular cardinals with \(\lambda \geq \kappa^{+3}\). Then there is a forcing notion \({\mathbb{P}}\) such that \(\vdash_{{\mathbb{P}}}2^{\mu}=\lambda\) and \(\lambda\) \(\to (\lambda,[\kappa;\kappa])\). \({\mathbb{P}}\) adds \(\lambda\) Cohen subsets of \(\mu\) in a special way and the fact that \(\lambda \to (\kappa^+)^ 3_{\kappa}\) is used to obtain the partition relation in \(V^{{\mathbb{P}}}.\) 2. Starting with the Erdős cardinal \(\kappa (\omega_ 1)\), i.e. the minimal \(\kappa\) with \(\kappa \to (\omega_ 1)_ 2^{<\omega}\), the author proves \(2^{\aleph_ 0}\to [\omega_ 1]^ 2_ 3\) consistent. This shows that Sierpiński's \(2^{\aleph_ 0}\nrightarrow [\omega_ 1]^ 2_ 2\) is sharp. 3. If \(\lambda >\aleph_ 0\) is regular and contains a stationary S such that \(S\cap \kappa\) is not stationary for any \(\kappa <\lambda\) then \(\lambda \nrightarrow [\lambda]^ 2_{\lambda}\). This generalizes Todorcevic's result and the proof resembles his proof of \(\aleph_ 1\nrightarrow [\aleph_ 1]^ 2_{\aleph_ 1}\).
scientific article; zbMATH DE number 4072949