Diamonds on trees

Abstract

We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $\omega_1$-tree, then $\diamondsuit_T \implies \diamondsuit$, and if $T$ is a Suslin tree, then $\diamondsuit_T \iff \diamondsuit$. We also prove that $\diamondsuit^*$ implies $\diamondsuit_T$ (yielding the consistency of $\diamondsuit_T$) and establish the consistency of $\neg\diamondsuit^* + (\forall T\text{ nonspecial }\omega_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $\omega_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $\sigma(S)$-closed and strategically closed in models -- which are preserved under countable support iterations.