#paper #tree #aronszajn #square # Abstract We characterize the tree of functions with ÿnite support in terms of definability. This turns out to have various applications: a new kind of tree dichotomy for $\omega_{1}$ on the one hand. On the other hand, we prove a re ection principle for trees on $\omega_{2}$ under SPFA. This refection of trees implies stationary refection. # 2. A class of minimal trees ## 2.1 Introducing coherence Def. $T$ is a normal tree in $2^{<\delta}$![[Trees#^trivcoh]] *Remark 2.2.* any normal tree with cofinal branches contains the full trivially coherent tree. **Lemma 2.4** - characteriztion of Trivially coherent trees # 3. Coherent Aronszajn-trees Def: ![[Trees#^uni-coh]] ![[Trees#^coherent]] Remark 3.1. Coherent trees with branches are trivially coherent. **Corollary 3.3.** Trivially coherent trees of regular height must have a cofinal branch. ## 3.2. An axiomatic approach to coherent Aronszajn-trees ![[Trees#^strong-homo]] **Lemma 3.2.** Let $\kappa$ be regular uncountable. If $T(f_{\nu}\mid \nu<\kappa)$ has a co5nal branch then every stationary $B \subseteq T(f_{\nu}\mid \nu<\kappa)$ contains a stationary chain $B_{0} \subseteq B$. **Remark 3.5.** Every uniformly coherent tree is strongly homogeneous. ^b726e1 **Theorem 3.6.** Every strongly homogeneous tree is isomorphic to a uniformly coherent tree. ^2a6840 **Theorem 3.7** There is no sequence $\left( f_{\nu} \mid \nu\to \omega_{2} \right)$ such that $f_{\nu}:\nu\to\omega_{1}$ is $1-1$ and for every $\nu<\mu$, $f_{\nu}=^{*}f_{\mu}\restriction \nu$. (i.e. you can't construct coherent trees on $\omega_{2}$ using injections when coherence means "up to a finite change") ## Constructing coherent Aronszajn trees **Theorem 3.9.** If $\square(\kappa)$ ([[Square principles#^bracket|Square bracket]]) holds, then there exists a uniformly coherent $𝜅$-Aronszajn tree. *Proof sketch.* Let $\left< C_{\nu} \mid \nu\in \mathrm{acc}(\kappa) \right>$ be a $\square(\kappa)$-sequence with $\min(C_{\nu})=1$. We construct inductively functions $f_{\nu}:\nu\to \nu$ such that 1. for every $\nu<\mu$, $f_{\nu}=^{*}f_{\mu}\restriction \nu$, 2. all $f_{\nu}s are almost 1-1, i.e. if $f_{\nu}(\alpha)=f_{\nu}(\beta)\ne 0$ then $\alpha=\beta$, 3. If $\gamma,\delta\in C_{\nu}$ and $\delta$ is the successor of $\gamma$ in $C_{\nu}$ then $f_{\nu}(\delta)=\gamma$, 4. $\lambda\in \mathrm{acc}(C_{\nu}) \iff f_{\lambda}=f_{\nu}\restriction \lambda$. Now, 4. together with the property of the $\square$ implies that there is no unbounded $D\subseteq \kappa$ such that for forall $\gamma<\delta$ from $D$, $f_{\gamma} \subseteq f_{\delta}$. This implies by lemma 3.2 that there is no cofinal branch in the tree $T(f_{\nu}\mid \nu<\kappa)$ (a branch somewhere in the tree can be moved to a branch in the trunk). This is a uniformly coherent tree by definition. **Corollary 3.11.** If $\square(\kappa)$ ([[Square principles#^bracket|Square bracket]]) holds, then there exists a uniformly coherent $𝜅$-Aronszajn tree of the for $g_{\alpha}:\alpha\to 2$. **Corollary 3.10.** The following are equiconsistent under ZFC: (1) there is a weakly compact cardinal, (2) every $\omega_{2}$-Aronszajn-tree does contain either an $\omega_{1}$-Aronszajn-subtree or a Cantor-subtree. ## The associated tree of a $\square$ sequence **Defnition 3.12.** If $\vec{C}= \left< C_{\nu} \mid \nu\in \mathrm{acc}(\kappa) \right>$ is a $\square(\kappa)$-sequence we deine the associated tree $(\kappa,<^{2})$ by letting $ \alpha<^{2}\beta \iff \alpha\in C'_{\beta}. $ In this context, $\vec{C}$ is called special if the associated tree $(\kappa,<^{2})$ is special. **Lemma 3.13.** $\square_{\lambda}$ is equivalent to the existence of a special $\square(\lambda^{+})$-sequence. **Theorem 3.14.** In the construction of Theorem 3.9, the following are equivalent: (1) $T(f_{\nu}\mid \nu<\kappa)$ is special, (2) $\left< C_{\nu} \mid \nu<\kappa \right>$ is special. **Corollary 3.15.** (a) If $\square_{\lambda}$ holds, there is a special coherent $\lambda ^{+}$-Aronszajn-tree. (b) If $\lambda\geq \omega_{2}$ and $\square(\lambda)$ holds, there is a non-special coherent $\lambda$-Aronszajn-tree.