Diamonds on trees
Osvaldo Guzmán, Carlos López-Callejas
TL;DR
This work generalizes Jensen’s diamond to ω1-trees via Brodsky’s tree-stationarity, introducing the tree- diamond principles $\diamondsuit_T$ and $\diamondsuit_T^*$. It establishes tight connections to the classical diamond: $\diamondsuit_T$ implies $\diamondsuit$ for nonspecial trees, and in almost-Suslin trees $\diamondsuit_T$ is equivalent to $\diamondsuit$; it also shows $\diamondsuit^*$ implies $\diamondsuit_T$ under broad conditions. The authors construct forcing models to separate these principles, proving that $\diamondsuit^*$ can fail while $\diamondsuit_T$ holds for all nonspecial Aronszajn trees, and that $\diamondsuit$ can hold while some tree $T$ violates $\diamondsuit_T$, thereby illustrating a nuanced landscape of tree-based diamonds. They further show that $\diamondsuit_T$ can be strictly stronger than $\diamondsuit$ in general and pose several open questions about successor partitions, ladder-system forcings, and extensions to larger cardinals. Overall, the paper develops a framework in which tree-based diamond principles serve as invariants for analyzing the structure of nonspecial ω1-trees and their forcing-independence properties.
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. \emph{The Bulletin of Symbolic Logic}]. In particular, we show that if $T$ is a nonspecial $ω_1$-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 $\neg\diamondsuit^* + (\forall T\text{ nonspecial }ω_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $ω_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $σ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations.