\(\mathbb P_{\max}\) variations for separating club guessing principles (Q2892674)

Let \({\mathcal L}\) denote the set of all countable infinite limit ordinals. The club weak club guessing principle (respectively, the interval hitting principle) asserts the existence of a sequence \(\langle a_\alpha : \alpha \in {\mathcal L}\rangle\) such that (i) \(a_\alpha\) is a cofinal subset of \(\alpha\) of order-type at most \(\omega\), and (ii) for any club \(D\subseteq\omega_1\), \(a_\alpha \cap D\) is infinite for club many \(\alpha\) (respectively, there is \(\alpha\in {\mathcal L}\) such that \(\{ \xi\in D : \beta\leq\xi < \min (a_\alpha \setminus (\beta +1))\} \not= \emptyset\) for all but finitely many \(\beta\in a_\alpha)\). The principle \((+)_{<\omega}^c\) asserts the existence of a sequence \(\langle F_\alpha : \alpha \in {\mathcal L} \rangle\) such that (i) \(F_\alpha\) is a family of club subsets of \(\alpha\) which is closed under finite intersections, and (ii) for any club \(D\subseteq\omega_1\), \(D\cap \alpha\in F_\alpha\) for club many \(\alpha\). The authors produce, from \(\mathrm {AD}^{L(\mathbb R)}\), a model in which the club weak club guessing principle and \((+)_{<\omega}^c\) both hold but the interval hitting principle fails.