Diamonds in mice

RESUMO

Let $M$ be a $(0,\omega_1+1)$-iterable mouse with no largest cardinal. Let $\gamma\leq\kappa$ be uncountable cardinals of $M$, with $\kappa$ regular in $M$. Then $M$ satisfies $\diamondsuit_{\kappa\gamma}^+$, and if $\kappa$ is non-ineffable in $M$ then $M$ satisfies $\diamondsuit_{\kappa\kappa}^+$. Suppose that either $\omega_1^M=\omega_1$ or $M$ satisfies "I am $(0,\omega_1+1)$-iterable". Let $\kappa\geq\omega_1^M$ be a cardinal of $M$. Then $M$ satisfies $\diamondsuit^*_{\kappa\omega_1^M}$.