#square #forcing_axioms Let $\kappa$ be a cardinal and $(P,\leq)$ be a partial order. $FA_\kappa(P)\equiv$ For all families $\{D_\alpha:\alpha<\kappa\}$ of predense subsets of $P$, there is a filter $G$ on $P$ meeting all these predense sets. Given a class $\Gamma$ of partial orders $FA_\kappa(\Gamma)$ holds if $FA_\kappa(P)$ holds for all $P\in \Gamma$. # AC/DC [[Viale - Useful axioms]] **Theorem.** $\mathsf{DC}_\kappa$ is equivalent to $FA_{\kappa}(<\kappa\text{-closed})$ over the theory $\mathsf{ZF}+\forall\lambda<\kappa\,\mathsf{DC}_\lambda$. ^76be71 # Diamond [[Velleman - Morasses, diamond, and forcing]] **Theorem.** $\diamondsuit_{\kappa}$ is equivalent to: Let $\mathbb{P} = (P, \leq )$ a poset and $\mathcal{D}=\{ D_{\alpha}\mid \alpha<\kappa \}$ a family of dense open sets. Suppose: 1. $\forall \alpha<\kappa$ $P_{\alpha}$ is lt;\kappa$-closed and $P=\bigcup_{\alpha<\kappa}P_{\alpha}$. 2. For each $\mathcal{L}$-structure $\mathfrak{B}$ with universe an ordinal $\alpha<\kappa$ we have a set $F_{\mathfrak{B}}\subseteq P$ such that $\{p \in P \mid \mathrm{rlm}(p) = \alpha\} \subseteq \mathrm{dns}(F_{\mathfrak{B}})[$. Then there is a set $G$ which is $\mathbb{P}$-generic over $\mathcal{D}$, and for any $\mathcal{L}$-structure $\mathfrak{A}$ with universe $\kappa$, $\exists \mathfrak{B}\prec_O \mathfrak{A}$ s.t. $G\cap F_{\mathfrak{B}}\ne \varnothing$. Furthermore, $G$ can be chosen to be $\kappa$-complete. # Squares >[!reminder]- >![[Velleman - On a generalization of jensen's □κ, and strategic closure of partial orders#^3a95d9]] [[Velleman - On a generalization of jensen's □κ, and strategic closure of partial orders]] **Thm** Let $κ$ be an infinite cardinal. TFAE 1. $B(\kappa)$ 2. For every poset $\mathbb{P}$, if GOOD has a winning strategy for $G_{\alpha}^{\mathbb{P}}$ for each $\alpha<\kappa$ then GOOD has a winning strategy for $G_{\kappa}^{\mathbb{P}}$. IE for every $\sigma$-closed poset, lt;\kappa$ strategic closure $\implies$ $\kappa$ strategic closure. 3. For every poset $\mathbb{P}$ and family of dense sets $\mathcal{D}$, if GOOD has a winning strategy for $G_{\alpha}^{\mathbb{P}}$ for each $\alpha<\kappa$ and $\lvert \mathcal{D} \rvert=\kappa$ then there is a $\mathbb{P}$-generic over $\mathcal{D}$. IE $FA_{\kappa}(\sigma\text{-closed and }<\kappa\text{ strategically closed})$ [[Ishiu, Yoshinobu - Directive trees and games on posets]] **Theorem.** Let $κ$ be an infinite cardinal. Then the following are equivalent: 1. $\square_{κ}=B(\kappa^{+})$ . 2. There is a continuous $(κ^{+} , κ)$-directive tree. 3. Every $(κ + 1)$-strategically closed poset is $κ^{+}$-strategically closed. 4. $FA_{\kappa ^{+}}(\kappa+1\text{ strategically closed})$ 5. Every $\sigma$-closed and $(κ + 1)$-strategically closed poset is $κ^{+}$-strategically closed. [[Yoshinobu - Approachability and games on posets]] **Theorem.** Let $\lambda$ be a regular uncountable cardinal. Then TFAE 1. $B(\lambda)$; 2. There exists a continuous $\lambda$-directive tree; 3. Every poset which is $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is $\lambda$-strategically closed. 4. $FA_{\lambda}(<\lambda\text{ strategically closed})$ >[!reminder]- >![[Yoshinobu - Approachability and games on posets#^3a915a]] # Approachability >[!reminder]- >![[Square principles#AP Definition]] >![[Yoshinobu - Approachability and games on posets#A generalization of AP]] [[Yoshinobu - Approachability and games on posets]] **Theorem.** Let $\lambda$ be a regular uncountable cardinal. Then TFAE 1. $A(\lambda)$; 2. There exists a $\lambda$-directive tree; 3. Every poset which is strongly $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is strongly $\lambda$-strategically closed. 4. Every poset which is strongly $(\kappa+1)$-strategically closed for every cardinal $\kappa<\lambda$ is $\lambda$-strategically closed. In particular, $\mathrm{AP}_{\kappa}\iff$Every strongly $(\kappa+1)$-strategically closed poset strongly $\kappa^{+}$-strategically closed.