Diamond principles and Tukey-top ultrafilters on a countable set

Abstract

We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_{\lambda}(U), \ \diamondsuit^p_\lambda(U)$) on $\omega$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These constructions are essentially different from Isbell's construction \cite{Isbell65} of Tukey-top ultrafilters. We prove using the Borel-Cantelli Lemma that full guessing is not possible and rule out several stronger guessing principles e.g. we prove that no Dodd-sound ultrafilters exist on $\omega$. We then apply these guessing principles to force a $q$-point which is Tukey-top (answering a question from \cite{Benhanou/Dobrinen23}), and prove that the class of ultrafilters which satisfy $\neg\diamondsuit^{-}_\lambda$ is closed under Fubini sum. Finally, we show that $\diamondsuit^{-}_\lambda$ and $\diamondsuit^p_\lambda$ can be separated.