# Definitions ## Basic definitions Let $T$ be a set and lt;_{T}$ an order relation on $T$. - The *downward cone* of $x\in T$ is $x_{\downarrow}=\{ y\in T \mid y<_{T}x\}$. - The *upward cone* of $x\in T$ is $x^{\uparrow}=\{ y\in T \mid y<_{T}x\}$. **Remark.** This is also commonly denoted $T_{x}$. - $(T,<_{T})$ is a *tree* if for every $x \in T$, $x_{\downarrow}$ is well-ordered. **Remark.** It is common refer to $T$ itself as the tree. - Denote $x \parallel y$ ($x,y$ are *compatible*) iff $x \leq_{T} y \lor y \leq_{T}x$ - Denote $x \bot y$ ($x,y$ are *incompatible*) iff $\neg x \parallel y$ - $A \subseteq T$ is a *chain* if $\forall x,y \in A(x \parallel y)$. - A *branch* is a cofinal chain. - $A \subseteq T$ is an *antichain* if $\forall x,y \in A(x \bot y)$ - The *subtree generated by $x \in T$* is $T^{x}=\{ t\in T \mid t \parallel x\}$. **Remark.** The notation for this term varies in the literature. - The *height of $x\in T$* is $\mathrm{ht}(x)=\mathrm{otp}(x_{\downarrow},<_{T})$ **Remark.** This is sometimes called the *rank* of $x$. - The $\alpha$th *level* of $T$ is $T_{\alpha}=\{ x\in T \mid \mathrm{ht}(x)=\alpha \}$. **Remark.** The notation for this term varies in the literature. - The *height of $T$* $\mathrm{ht}(T)$ is the first $\alpha$ such that $T_{\alpha}=\varnothing$. Equivalently, $\mathrm{ht}(T)=\sup_{x\in T} \mathrm{ht}(x)$. **Remark.** This is sometimes called the *rank* of $T$. - A tree is *rooted* if $|T_{0}|=1$ (in which case this single element is the *root*). - $D\subseteq T$ is *dense* if $T=\bigcup_{x\in D}x_{\downarrow}$. I.e. $\forall t\in T \exists x\in D (t \leq_{T} x)$. - $D\subseteq T$ is *open* if $\forall x\in D$, $x^{\uparrow}\subseteq D$. I.e. $\forall x\in D (x<y \to y\in D)$. ## Types of trees ### Basic properties - A tree is *Normal* if $\forall x\in T$, $\forall \alpha<\mathrm{ht}(T)$ there is a node in $T_{\alpha}$ compatible with $x$. ^normal - A tree is *Hausdorff* if for every limit $\alpha$ and $s,t \in T_{\alpha}$, $s_{\downarrow}=t_{\downarrow} \to s=t$. ^Hausdorff - *A $\kappa$-tree* is a tree of height $\kappa$ with all levels of size lt;\kappa$.^kappa-tree ### Branch and antichain properties - A *$\kappa$-Aronszajn* tree is a $\kappa$-tree where all chains are of size lt;\kappa$. ^aron - A *$\kappa$-Suslin* tree is a $\kappa$-tree where all antichains (and thus also chains) are of size lt;\kappa$. ^Suslin - A *$\kappa$-almost-Suslin* tree is a $\kappa$-Aronszajn tree where all antichains are non-stationary (i.e. the levels intersecting the antichain are non-stationary). ^almost-Suslin - A $\lambda^{+}$-tree is *special* if it may be covered by $\lambda$ many antichains ^spec - A normal $\lambda^{+}$-Aronszajn tree is *$\lambda$-distributive* if every intersection of $\lambda$ many dense open subsets is dense. ^dist - A *$\kappa$-Kurepa* tree is a $\kappa$-tree with more than $\kappa$ many chains of size $\kappa$. ^kurepa > [!info] Definition > $\kappa$ has the *tree property* if there are no $\kappa$-Aronszajn trees. > ^tree-prop ### Homogeneity properties - A tree $T$ is said to be *homogeneous* provided $x^↑$ and $y^↑$ are isomorphic for every $x, y ∈ T$ with $ht(x) = ht(y)$. ^homogeneous - A tree $T$ is said to be *strongly homogeneous* If there is a family $\{ h_{x,y} \mid x,y \in T, \mathrm{ht}(x)=\mathrm{ht}(y) \}$ of automorphisms with the following conditions: 1. $h_{x,y}$ moves $T^{x}$ to $T^{y}$ and vice versa, so $x$ is mapped to $y$, and is the identity in all other parts of the tree. $h_{x,x}$ is the identity on $T$. 2. (commutativity) $h_{x,z}(t)=h_{y,z}(h_{x,y}(t))$ forall $x,y,z \in T_{\alpha}$ and $t \geq x$. 3. (uniformity) If $x,y,x',y'$ are such that $x'\geq x$ and $h_{x,y}(x')=y'\geq y$ then $h_{x',y'}\restriction T^{x'} = h_{x,y}\restriction T^{x'}$ 4. (transitivity) If $\alpha$ is limit and $x,y \in T_{\alpha}$ then there are $z,w \in T_{<\alpha}$ such that $h_{z,w}(x)=y$. ^strong-homo ![[König - Local coherence#^2a6840]] ^2005a8 > [!remark] > For any infinite cardinal $𝜆$, and any normal $𝜆^+$-Aronszajn tree 𝒯 : > 𝒯 is $𝜆^+$-Suslin ⇒ 𝒯 is $𝜆$-distributive ⇒ 𝒯 is not special. ## Sequential/Streamlined trees ^9d99dc $(T,<_{T})$ is called a *sequential tree* (see [[Gaifman, Specker - Isomorphism types of trees#^sequential|Gaifman and Specker]]) if $T$ is a set of sequences such that for all $t\in T$ and $\beta<\mathrm{dom}(t)$, $t\restriction\beta\in T$, and lt;_{T}=\subset$.^seque >[!note] Remark >A sequential tree is always rooted and Hausdorff. Conversly every rooted Hausdorff tree is isomorphic to a sequential tree, by associating each $t$ with the enumeration of $t_{\downarrow}$. $T$ is called a *streamlined tree* (see [[Brodsky, Rinot - A microscopic approach to Souslin-tree construction, Part II|Brodsky and Rinot]]) iff there exists some cardinal $\kappa$ such that $T\subseteq{}^{<\kappa}H_\kappa$ and, for all $t\in T$ and $\beta<\mathrm{dom}(t)$, $t\restriction\beta\in T$. ^stream >[!note] Remark >Formally every sequential tree is a streamlined tree for some large enough $\kappa$, but when thinking of streamlined trees $T\subseteq{}^{<\kappa}H_\kappa$ we usually mean that the height of $T$ is $\kappa$. *All subsequent trees are normal*. A streamlined tree $T$ is called: - *Trivially coherent*: $\forall s\in T$, the support of $s$ is finite (where $supp(s)=\{ \alpha \in dom(s) \mid s(\alpha)\ne 0\}$) ^trivcoh - *Coherent*: $\forall s,t\in T$ the set $\{ \alpha\mid s(\alpha)\ne t(\alpha) \}$ is finite. Equivalently: A normal subtree of a uniformly corerent tree. ^coherent For $s,t$ with $\mathrm{dom}(s)=\alpha<\beta=\mathrm{dom}(t)$ define $s*t=\begin{cases} s(i) & i<\alpha \\ t(i) & \alpha\leq i <\beta \end{cases}$ - *Uniformly homogeneous*: $\forall s\in T_{\alpha},t\in T_{\beta}$, $s*t\in T_{\beta}$ ^uni-homo - *Uniformly coherent*: Coherent and uniformly homogeneous. Equivalently: There is a coherent sequence $f_{\alpha}:\alpha\to\alpha$ such that $ T = T(f_{\alpha}\mid \alpha<\kappa) = \{ f: \alpha \to \alpha \mid f=^{*}f_{\alpha}, \alpha<\kappa\} $^uni-coh ![[König - Local coherence#^b726e1]] # General facts For an inaccessible $\kappa$, non-existence of $\kappa$-Aronszajn trees is an equivalent definition of [[Weakly compact cardinal]]. **Fact.** (see Kunen pg. 225) Let $\kappa$ be strongly inaccessible cardinal, and assume that there are no $\kappa$-Aronszajn trees. Prove that whenever $S\subseteq \kappa$ is stationary, there is an uncountable regular $\lambda<\kappa$ such that $S\cap \lambda$ is stationary in $\lambda$. In particular, $\kappa$ is strongly Mahlo, and not the first strongly Mahlo cardinal. **Fact.** (see Kunen pg. 217) GCH implies there is a $\kappa$-Aronszajn tree for every $\kappa$ which is the successor of a regular. ^4605f4 # Trees and [[Square principles]] ## Aronszajn trees **Fact** (cited in [[Brodsky, Rinot - Distributive Aronszajn trees]]). For every infinite cardinal 𝜆: 1. (Jensen, [Jen72]) $\square ^{*}_{\lambda}=\square_{\lambda}(\lambda^{+},<\lambda ^{+})$ holds iff there exists a special $\lambda ^{+}$-Aronszajn tree; 2. (Ben-David and Shelah, [BS86]) If $𝜆$ is a singular strong-limit cardinal and $2^{\lambda}=\lambda ^{+}$ , then $\square ^{*}_{\lambda}=\square_{\lambda}(\lambda^{+},<\lambda ^{+})$ entails the existence of a normal $𝜆$-distributive $𝜆^+$-Aronszajn tree. For every regular uncountable cardinal 𝜅: 3. (Todorcevic, [Tod87]) $\square(𝜅, <𝜅)$ holds iff there exists a $𝜅$-Aronszajn tree; 4. ([[König - Local coherence]]) If $\square(\kappa)=\square(𝜅, <2)$ holds, then there exists a uniformly coherent $𝜅$-Aronszajn tree. 5. (Brodsky, Rinot) Assume GCH, and that $𝜆$ is some singular cardinal. If either $\square_{\lambda}(\lambda^{+},<\lambda^{+})$ or $\square(\lambda^{+},<\lambda)$ holds, then there exists a normal $𝜆$-distributive $𝜆^+$-Aronszajn tree. ## Special trees 1. ([[König - Local coherence]]) If $\square_{\lambda}$ holds, there is a special coherent $\lambda ^{+}$-Aronszajn-tree. 2. (Todorcevic, [Tod89]) If $\lambda\geq \omega_{2}$ and $\square(\lambda)$ holds, there is a non-special coherent $\lambda$-Aronszajn-tree. ## Suslin trees **Suslin's hypothesis** states that there are no Suslin trees. ^dde5ed **Fact** [[Brodsky, Rinot - Distributive Aronszajn trees]]. (Rinot implicit in [Rin17]). Assume GCH, $\kappa$ is regular uncountable cardinal. If $\square(𝜅^{+} , <𝜅)$ holds, then there exists a $\kappa ^{+}$-Suslin tree. ![[Brodsky, Rinot - A microscopic approach to Souslin-tree construction, Part II#^aa32f8]]