Diamond principle ($♢$)

A sequence $D_{\alpha }\subseteq \alpha$ guesses every $X\subseteq {\omega }_1$ stationarily often: $\{\alpha :D_{\alpha }=X\cap \alpha \}$ is stationary.

For each index, collect the ground subsets which some condition forces to be the corresponding diamond guess. Distinct forced values have incompatible witnessing conditions, so the countable chain condition for forcing makes this a countable family. Every ground subset is guessed by these families on a ground stationary set, by testing each ground club set. The countable-family diamond equivalence then yields diamond in the ground model. A generic guess need not itself be a ground subset; only values forced equal to one are collected.

For a stationary $S\subseteq {\omega }_1$, a sequence $D_{\alpha }\subseteq \alpha$ indexed by $S$ guesses every $X\subseteq {\omega }_1$ on a stationary subset of $S$.

A sequence $D_{\alpha }\subseteq \alpha$, indexed by a stationary subset $S$ of a regular uncountable cardinal $\lambda$, guesses every $X\subseteq \lambda$ stationarily often: $\{\alpha \in S:D_{\alpha }=X\cap \alpha \}$ is stationary. This extends the usual stationary diamond principle at ${\omega }_1$.

At each $\alpha \in S$, a countable family $E_{\alpha }\subseteq \mathcal{P}(\alpha )$ is supplied. Every $X\subseteq \lambda$ has $X\cap \alpha \in E_{\alpha }$ on a stationary subset of $S$. The countable-family diamond equivalence shows that this apparently weaker prediction is equivalent to a single diamond guess at each index.

Enumerate each countable guessing family and use a bijection $\pi :\lambda \times \omega \to \lambda$ with a club set of prefix-closure points. The $n$th candidate sequence decodes the $n$th component of the $n$th family entry. If all candidates fail, choose counterexample sets $X_n$ and club sets witnessing failure, then code the $X_n$ together using $\pi$. A correct family guess on the common closure club set and all failure club sets decodes to one of the forbidden correct guesses. The singleton-family implication supplies the reverse direction.

At a correctly guessing limit stage of a normal tree construction, make every new level node extend a member of the guessed maximal tree antichain. It follows that the antichain has no later member. This produces a Suslin tree from the stationary diamond principle.