#definition #combinatorial_principle **Def** For a regular $\kappa$, $\diamondsuit_{\kappa}=\diamondsuit(\kappa)$ is the assertion that there is a sequence $\left< S_{\alpha}\mid \alpha<\kappa \right>$ such that $\forall \alpha< \kappa(S_{\alpha}\subseteq \alpha)$ and $\forall X\subseteq \kappa$ $(\left\{ \alpha \mid X\cap \alpha=S_{\alpha} \right\}$ is stationary in $\kappa$ ). ^99868e **Fact** The poset of partial functions from $\kappa$ to $\kappa$ of size lt;\kappa$, adds a $\diamondsuit_{\kappa}$-sequence. **Eq. Def** $\diamondsuit^{-}(H_{\kappa})$ asserts the existence of a sequence $\left< S_{\alpha}\mid \alpha<\kappa \right>$ such that for every parameter $p ∈ H(κ^{+})$ and subset $S ⊆ H(κ)$, there exists an elementary submodel $\mathcal{M} ≺ H(κ^{+})$ containing $p$ such that $κ^{\mathcal{M}} := M ∩ κ$ is an ordinal lt; κ$ and $S_{\kappa^{\mathcal{M}}} = M ∩ S$. See [[Brodsky, Rinot - A Microscopic approach to Souslin-tree constructions. Part I]] ^ac6db4 # Generalizations [[Kunen - Set theory]] pg 232 **Def** Let $\kappa$ be regular uncountable, $S$ a stationary subset of $\kappa$. $\left< A_{\alpha}\mid \alpha \in S\right>$ such that $A_{\alpha}\subseteq \alpha$ is a $\diamondsuit_{S}(\kappa)$ sequence if $\forall X\subseteq \kappa$ $\left\{ \alpha \in S \mid X\cap \alpha=A_{\alpha} \right\}$ is stationary in $\kappa$. $\left< \mathcal{A}_{\alpha}\mid \alpha \in S\right>$ such that $\mathcal{A}_{\alpha}\subseteq \mathcal{P}(\alpha)$ and $\lvert \mathcal{A}_{\alpha} \rvert \leq \lvert \alpha \rvert$ is a - $\diamondsuit^{-}_{S}(\kappa)$ sequence $\forall X\subseteq \kappa$ $\left\{ \alpha \in S \mid X\cap \alpha\in \mathcal{A}_{\alpha} \right\}$ is stationary in $\kappa$. - $\diamondsuit^{*}_{S}(\kappa)$ sequence if $\forall X\subseteq \kappa$ there is a club $C\subseteq \kappa$ such that $\forall \alpha \in C\cap S$ $X\cap \alpha \in \mathcal{A}_{\alpha}$. - $\diamondsuit^{+}_{S}(\kappa)$ sequence if $\forall X\subseteq \kappa$ there is a club $C\subseteq \kappa$ such that $\forall \alpha \in C\cap S$ $X\cap \alpha \in \mathcal{A}_{\alpha}$ and $C\cap \alpha \in \mathcal{A}_{\alpha}$. **Fact** $ \diamondsuit^{+}(\kappa)\to \diamondsuit^{*}(\kappa)\to \diamondsuit^{-}(\kappa)\leftrightarrow \diamondsuit(\kappa)\to 2^{<\kappa}=\kappa $