Some results about (+) proved by iterated forcing (Q2892673)

Let \({\mathcal L}\) denote the set of all countable infinite limit ordinals. A club guessing (respectively, strong club guessing) sequence on \(\omega_1\) is a sequence \(\langle C_\delta : \delta\in {\mathcal L} \rangle\) such that (i) \(C_\delta\) is a cofinal subset of \(\delta\), and (ii) for any club \(D\subseteq \omega_1\), the set of all \(\delta\in {\mathcal L}\) such that \(C_\delta\setminus C\) is finite is cofinal in \(\omega_1\) (respectively, contains a club subset of \(\omega_1\)).NEWLINENEWLINEThe principle \((+)\) asserts the existence of a sequence \(\langle F_\delta : \delta\in {\mathcal L} \rangle\) such that (i) \(F_\delta\) is a family of subsets of \(\delta\) which is closed under supersets, contains all cobounded subsets of \(\delta\) and is such that the intersection of any two of its members is cofinal in \(\delta\), and (ii) for any club \(D\subseteq\omega_1\), \(\{ \delta\in {\mathcal L} : D \cap\delta \in F_\delta\} \not= \emptyset\).NEWLINENEWLINEThe authors prove that the following three statements are consistent: (A) GCH + the negation of (+); (B) CH + (+) + there is no guessing sequence on \(\omega_1\); and (C) \(\diamondsuit^+\) + there is no strong club guessing sequence on \(\omega_1\).