[Assaf Rinot's Blog](https://blog.assafrinot.com/?p=3687) # Classical squares See [[Jensen - The fine structure of the constructible hierarchy]] **Def** For any class of ordinals $S$, a *coherent sequence of clubs on $S$* is $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha\text{ a singular ordinal}\right\rangle$ such that: 1. $C_{\alpha}$ is club in $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$. A square on $S$ is a coherent sequence s.t. for every $\alpha \in S$ 3. $\mathrm{otp}(C_{\alpha})<\alpha$ ## Jensen's Global square (on singulars) ^34923d **Def** The *global square principle* asserts the existence of a class sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha\text{ a singular ordinal}\right\rangle$ such that: 1. $C_{\alpha}$ is club in $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$; 3. $\mathrm{otp}(C_{\alpha})<\alpha$. **Def** For an ordinal $\theta$ we say that $\square_{(\theta)}$ holds if there is a sequence $ \mathcal{C}=\left\langle C_{\alpha}\mid\alpha<\theta\text{ a singular limit ordinal}\right\rangle $ satisfying (1-3) above (this corresponds to the principle $\square_{(0,\theta)}$ in definition 1.5 of [[Džamonja, Shelah - Saturated filters at successors of singulars, weak reflection and yet another weak club principle]]). We call such a sequence a $\square_{(\theta)}$-sequence. ## Jensen's Square for successors $\square_{\kappa}$ ### Definition **Def** For a regular cardinal $\kappa$, the principle $\square_{\kappa}$ asserts the existence of a sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha\in\mathrm{acc}(\kappa^{+})\right\rangle$ 1. $C_{\alpha}$ is club in $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$; 3. If $\mathrm{cf}(\alpha)<\kappa$ then $\left|C_{\alpha}\right|<\kappa$. **Remark.** It follows that If $\mathrm{cf}(\alpha)=\kappa$ then $\mathrm{otp}(C_{\alpha})=\kappa$. Hence we can use the following: **Equivalent Def** For a regular cardinal $\kappa$, the principle $\square_{\kappa}$ asserts the existence of a sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha\in\mathrm{acc}(\kappa^{+})\right\rangle$ 1. $C_{\alpha}$ is club in $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$; 3. $\mathrm{otp}(C_{\alpha})\leq\kappa$. ### Weak squares **Def.** $\square_{\kappa,\nu}$ asserts the existence of a sequence $\left< \mathcal{C}_{\alpha}\mid \alpha\in \mathrm{Lim}(\kappa ^{+} \right>$ such that for all $\alpha\in \mathrm{Lim}(\kappa ^{+})$: 1. $\mathcal{C}_{\alpha}$ is a nonempty collection of clubs in $\alpha$; 2. $\lvert \mathcal{C}_{\alpha} \rvert \leq\nu$; 3. If $C\in \mathcal{C}_{\alpha}$ and $\bar{\alpha}\in\mathrm{acc}(\mathcal{C}_{\alpha})$ , $C\cap\bar{\alpha}\in \mathcal{C}_{\bar{\alpha}}$; 4. If $\mathrm{cf}(\alpha)<\kappa$ then $\left|C\right|<\kappa$ for every $C\in \mathcal{C}_{\alpha}$. $\square_{\kappa,<\nu}$ is the same with $\leq \nu$ replaced by lt;\nu$. $\square ^{*}_{\kappa}:=\square_{\kappa,\kappa}$ is also called *the* weak square. ## Velleman's principle $B(\kappa)$ ![[Velleman - On a generalization of jensen's □κ, and strategic closure of partial orders#A generalization of $ square$]] ## Theorems ![[Ishiu, Yoshinobu - Directive trees and games on posets#^6bac72]] ![[Velleman - On a generalization of jensen's □κ, and strategic closure of partial orders#Other results]] ## Komjath's $A(\mu,\kappa)$ ![[Komjáth - The Colouring Number#^komjathweakbox]] # "Unthreadable" squares See [[Brodsky, Rinot - Distributive Aronszajn trees]] ### General definition >[!info] Definition > $\square_{\xi}(\kappa,<\mu)$ asserts the existence of a sequence $\left< \mathcal{C}_{\alpha}\mid \alpha<\kappa \right>$ such that for all limit $\alpha<\kappa$: > 1. $\mathcal{C}_{\alpha}$ is a nonempty collection of clubs in $\alpha$; > 2. $\lvert \mathcal{C}_{\alpha} \rvert <\mu$; > 3. For every $C\in \mathcal{C}_{\alpha}$ and $\bar{\alpha}\in\mathrm{acc}(C)$ , $C\cap\bar{\alpha}\in \mathcal{C}_{\bar{\alpha}}$; > 4. If $C\in \bigcup \mathcal{C}_{\alpha}$ then $\mathrm{otp}(C)\leq \xi$; > 5. There is no club $C$ in $\kappa$ such that for every $\bar{\alpha}\in\mathrm{acc}(C)$ , $C\cap\bar{\alpha}\in \mathcal{C}_{\bar{\alpha}}$. ^general-square >[!note] Remark > If $\xi=\kappa$ then 4. is redundant. > If $\xi<\kappa$ then 5. is redundant. ### Special cases - $\square(\kappa,<\mu):= \square_{\kappa}(\kappa,<\mu)$ - $\square(\kappa):=\square_{\kappa}(\kappa,<2)$: there is a sequence $\left< C_{\alpha}\mid \alpha<\kappa \right>$ such that for all limit $\alpha<\kappa$: ^bracket 1. $C_{\alpha}$ is club in $\alpha$, 2. For every $\bar{\alpha}\in\mathrm{acc}(C_{\alpha})$ , $C_{\alpha}\cap\bar{\alpha}= C_{\bar{\alpha}}$; 3. There is no club $C$ in $\kappa$ such that for every $\bar{\alpha}\in\mathrm{acc}(C)$ , $C\cap\bar{\alpha}=C_{\bar{\alpha}}$. - $\square_{\lambda}=\square_{\lambda}(\lambda^{+},<2)$ : there is a sequence $\left< C_{\alpha}\mid \alpha<\lambda ^{+} \right>$ such that for all limit $\alpha<\lambda ^{+}$:^subscript 1. $C_{\alpha}$ is club in $\alpha$, 2. For every $\bar{\alpha}\in\mathrm{acc}(C_{\alpha})$ , $C_{\alpha}\cap\bar{\alpha}= C_{\bar{\alpha}}$; 3. $\mathrm{otp}(C_{\alpha})\leq\lambda$ 4. There is no club $C$ in $\lambda ^{+}$ such that for every $\bar{\alpha}\in\mathrm{acc}(C)$, $C\cap\bar{\alpha}=C_{\alpha}$. (follows from 3) - $\square ^{*}_{\lambda}:=\square_{\lambda}(\lambda^{+},<\lambda ^{+})$ : there is a sequence $\left< \mathcal{C}_{\alpha}\mid \alpha<\lambda ^{+} \right>$ such that for all limit $\alpha<\lambda ^{+}$: ^star 1. $\mathcal{C}_{\alpha}$ is a nonempty collection of clubs in $\alpha$; 2. If $C\in \bigcup \mathcal{C}_{\alpha}$ then $\mathrm{otp}(C)\leq \lambda$; 3. $\lvert \mathcal{C}_{\alpha} \rvert <\lambda ^{+}$; 4. For every $C\in \mathcal{C}_{\alpha}$ and $\bar{\alpha}\in\mathrm{acc}(\mathcal{C}_{\alpha})$ , $C\cap\bar{\alpha}\in \mathcal{C}_{\bar{\alpha}}$; 5. There is no club $C$ in $\lambda ^{+}$ such that for every $\bar{\alpha}\in\mathrm{acc}(C)$ , $C\cap\bar{\alpha}\in \mathcal{C}_{\bar{\alpha}}$. ### Corollaries from Velleman >[!note] Remark > A $B(\eta,\tau)$ sequence is an $\square_{\xi}(\eta,<2)$ sequence for $\xi<\tau$. > A $\square_{\xi}(\eta,<2)$ sequence is a $B(\eta,\xi+1)$ sequence Velleman's results imply that for every cardinal $\kappa$ and every $\kappa+1<\tau<\kappa ^{+}$, $\kappa\leq\xi<\kappa ^{+}$, TFAE: 1. $B(\kappa ^{+})$ 2. $B(\kappa ^{+},\tau)$ 3. $\square_\xi(\kappa^{+},<2)$ See [[Trees#Trees and Square principles]] # Approachability Shelah, Foreman ### AP Definition For a regular cardinal $\kappa$, the principle $\mathrm{AP}_{\kappa}$ asserts the existence of a sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha<\kappa^{+}\right\rangle$ such that 1. $C_{\alpha}\subseteq\kappa^{+}$ and $\mathrm{otp}(C_{\alpha})\leq \kappa$; And there is a club subset $C$ of $\mathrm{acc}(\kappa^{+})$ such that for every $\alpha \in C$, 2. $C_{\alpha}$ is a club subset of $\alpha$; 3. $\forall\beta<\alpha \exists \gamma<\alpha$ such that $C_{\alpha}\cap\beta=C_{\gamma}$. ### Yoshinobu's principle ![[Yoshinobu - Approachability and games on posets#A generalization of AP]] ![[Yoshinobu - Approachability and games on posets#Main theorem]] # X-box ("hitting" square) [[Brodsky, Rinot - A Microscopic approach to Souslin-tree constructions. Part I]] ### Definition $\boxtimes^{-}$ For a regular uncountable cardinal $\kappa$ and any stationary $S \subseteq \kappa$, the principle $\boxtimes^{-}(S)$ asserts the existence of a sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha<\kappa\right\rangle$ such that 1. $C_{\alpha}$ is club in $\alpha$ for every limit $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$; 3. For every cofinal $A\subseteq \kappa$, there are stationarily many $\alpha\in S$ such that $\sup(\mathrm{nacc}(C_{\alpha})\cap A)=\alpha$. **Thm** If $V=L$ then $\boxtimes^{-}(\kappa)$ holds for every uncountable regular non-weakly-compact $\kappa$. ### Definition $\boxtimes$ **Def** For a regular uncountable cardinal $\kappa$ and any stationary $S \subseteq \kappa$, the principle $\boxtimes(S)$ asserts the existence of a sequence $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha<\kappa\right\rangle$ such that ^xbox 1. $C_{\alpha}$ is club in $\alpha$ for every limit $\alpha$; 2. If $\beta\in\mathrm{acc}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$; 3. For every sequence $\left< A_{i} \mid i<\kappa \right>$ of cofinal subsets of $\kappa$ , there are stationarily many $\alpha\in S$ such that $\sup \{ \beta<\alpha\mid \mathrm{succ}_{\omega}(C_{\alpha}\setminus \beta)\subseteq A_{\alpha} \}=\alpha$.