Generic compactness reformulated (Q701724)

The author introduces the Game Reflection Principle GRP\(^+\) that asserts that for every regular uncountable cardinal \(\lambda\), player II has a winning strategy in the two-person game \(G_\lambda\) that is defined as follows: The game lasts \(\omega_1\) moves, with player I moving first at limit steps. I produces a sequence \(\langle f_\alpha : \alpha < \omega_1\rangle\), where \(f_\alpha : [\lambda]^{\omega_1} \to \lambda\) and \(f_\alpha (a)\in a\) for all \(a\in [\lambda]^{\omega_1}\), and II produces a sequence \(\langle \delta_\alpha : \alpha < \omega_1\rangle\), where \(\delta_\alpha\in\lambda\). II wins if and only if \(\Big| \bigcap_{\alpha < \beta} f_\alpha^{-1} (\delta_\alpha )\Big| \geq 2\) for every \(\beta < \omega_1\). It is shown that GRP\(^+\) holds if and only if \(\omega_2\) is generically supercompact by \(\omega\)-closed forcing. It follows that if \(\kappa\) is a supercompact cardinal and \({\mathbb P}\) is the Levy collapse of every uncountable cardinal less than \(\kappa\) to \(\omega_1\), then, in \(V^{\mathbb P}\), GRP\(^+\) holds. It is also shown that CH, Rado's conjecture and the semiproperness of Namba forcing are all implied by GRP\(^+\). Reviewer's comment: Proposition 1 is due to the reviewer [see ``Concerning stationary subsets of \([\lambda]^{<\kappa}\)'', Lect. Notes Math. 1401, 119--127 (1989; Zbl 0683.03027)].