Diamonds on trees

TL;DR

This paper explores the relationships between the diamond principle and stationary trees, establishing new implications and consistency results for various types of trees and diamond principles.

Contribution

It generalizes the diamond principle for trees using stationarity, proving new implications and consistency results for nonspecial and Suslin trees.

Findings

If T is a nonspecial ω₁-tree, then iamondsuit_T implies iamondsuit.

For Suslin trees, iamondsuit_T is equivalent to iamondsuit.

The paper establishes the consistency of iamondsuit_T for all nonspecial -trees under certain conditions.

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.