More on weak diamond (Q2236637)

Let \(\lambda\) be a regular uncountable cardinal. For a cardinal \(\theta \geq 2\), weak diamond for \(\theta\) colours asserts the existence of \(F : \bigcup_{\alpha< \lambda} {}^{\alpha} 2 \rightarrow \theta\) and \(\eta : \lambda \rightarrow \theta\) with the property that the set \(\{\delta \in \lambda : \eta (\delta) = F (f \vert \delta) \}\) is stationary for all \(f :\lambda \rightarrow 2\). The author establishes the following dichotomy : Assuming that weak diamond holds for two colours, either weak diamond for \(\theta\) colours also holds, where \(\theta \geq 3\), or some ideal associated with \(F\) is not \(\lambda^+\)-saturated. An application to model theory is included. The paper is rather technical and at times difficult to follow. It is the reviewer's guess that in Conclusion 2.6 Id\(^b (F)\) should be Id\(^b (F^{\otimes})\).