\(\diamondsuit\) at Mahlo cardinals (Q2710608)

Let \(\kappa\) be a Mahlo cardinal. Given \(S\subseteq C_\kappa\), where \(C_\kappa\) denotes the set of all infinite cardinals \(<\kappa\), \(\diamondsuit_\kappa(S)\) asserts the existence of \(s_\mu\subseteq \mu\) for \(\mu\in S\) such that the set \(\{\mu\in S: s_\mu= A\cap \mu\}\) is stationary in \(\kappa\) for every \(A\subseteq\kappa\). Kunen proved that \(\diamondsuit_\kappa(\{\mu\in C_\kappa: \mu\) is regular\}) holds if \(\kappa\) is subtle. Extending a result of Woodin, Hauser showed that for any \(m,n> 0\), it is equiconsistent with the existence of a \(\Pi^m_n\)-indescribable cardinal to have a \(\kappa\) \(\Pi^m_n\)-indescribable so that \(\diamondsuit_\kappa(\{\mu\in C_\kappa:\mu\) is regular\}) fails. Jensen then established that the failure of \(\diamondsuit_\kappa(\{\mu\in C_\kappa: \text{cf}(\mu)= \omega_1\})\) implies the existence of \(0^\#\). Woodin finally proved the consistency of the failure of \(\diamondsuit_\kappa(C_\kappa)\) starting from an assumption that \(\kappa\) is hypermeasurable with order slightly more than \(\kappa^{++}\). The author proves the following two results: (A) Suppose there is no sharp for inner models with a strong cardinal. Then for every regular cardinal \(\nu\) with \(\omega_1< \nu< \kappa\), \(\diamondsuit_\kappa(\{\mu\in C_\kappa: \text{cf}(\mu)= \nu\})\) holds in \({\mathbf V}\) provided that there are only non-stationarily many \(\rho<\kappa\) with \(\text{o}(\rho)\geq \nu\) in \({\mathbf K}\). (B) Suppose there is no inner model for \(\text{o}(\tau)= \tau^{++}\). Then for every regular cardinal \(\nu\) with \(\omega_1\leq \nu<\kappa\), \(\diamondsuit_\kappa(\{\mu\in C_\kappa: \text{cf}(\mu)= \nu\})\) holds in \({\mathbf V}\) provided that there are only non-stationarily many ordinals \(<\kappa\) of cofinality \(\nu\) which become regular in \({\mathbf K}\).