On full Souslin trees

A microscopic approach to Souslin-tree construction, Part II

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of $\diamondsuit$. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path,…

Optimal Matrices of Partitions and an Application to Souslin Trees

The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result for strongly homogeneous Souslin trees. The latter is in turn applied to separate strong notions of rigidity of Souslin trees, thereby answering a considerable portion of a question of Fuchs and Hamkins.

A Microscopic approach to Souslin-tree constructions. Part I

We propose a parameterized proxy principle from which $κ$-Souslin trees with various additional features can be constructed, regardless of the identity of $κ$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $κ$-Souslin tree that applies also for $κ$ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that …

Preservation of a Souslin tree and side conditions

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.

Many countable support iterations of proper forcings preserve Souslin trees

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

Reduced powers of Souslin trees

We study the relationship between a $κ$-Souslin tree $T$ and its reduced powers $T^θ/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $θ$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $ T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.

小基数上的满 Souslin 树

Full Souslin trees at small cardinals

本文改进 Kunen 问题的已有结果，证明满 Souslin 树可在小基数上相容存在，并构造 $\aleph_3$ 棵满足任意可数棵乘积仍为 $\aleph_2$-Souslin 树的满 $\aleph_2$-树。

论Souslin树及其泛型分支的刚性

On the rigidity of Souslin trees and their generic branches

该数理逻辑研究证明了存在满足特定刚性与Kurepa性质的Souslin树的相容性，解答了Fuchs的相关问题，还证明了与菱形原理相容的Souslin树稠密子集性质，回应了Baumgartner的相关问题。

A forcing axiom deciding the generalized Souslin Hypothesis

We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.

The Ostaszewski square, and homogenous Souslin trees

Assume GCH and let $λ$ denote an uncountable cardinal. We prove that if $\square_λ$ holds, then this may be witnessed by a coherent sequence $< C_α| α< λ^+ >$ with the following remarkable guessing property: For every sequence $< A_i | i<λ>$ of unbounded subsets of $λ^+$, and every limit $θ<λ$, there exists some $α<λ^+$ such that $\otp(C_α)=θ$, and the $(i+1)_{th}$-element of $C_α$ is a member of $A_i$, for all $i<θ$. As an application, we construct an homogenous $λ^+$-Souslin tree from $GCH+\square_λ$, for every singular cardinal $λ$. In addition, as a by-product, a theorem of Farah and Velic…

Changing the Heights of Automorphism Towers by Forcing with Souslin Trees over L

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

A guessing principle from a Souslin tree, with applications to topology

We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces. Our main result states that strong instances of $\clubsuit_{AD}$ follow from the existence of a Souslin tree. It is also shown that the weakest instance of $\clubsuit_{AD}$ does not follow from the existence of an almost Souslin tree. As an application, we obtain a simple, de Caux type proof of Rudin's result…

组合集合论中的代理原理（Proxy principles）

Proxy principles in combinatorial set theory

本文全面阐述组合集合论中的参数化代理原理，梳理其研究进展与应用，并给出新结果：对任意正整数$n$，存在$\lambda^+$-Souslin树，其所有$n$-导出树为Souslin树但$(n+1)$-幂特殊。

On the ideal $J[κ]$

Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[κ]$, from which we confirm that $κ$-Souslin trees exist in various models of interest. As a corollary we get that for every integer $n$ such that $\mathfrak b<2^{\aleph_n}=\aleph_{n+1}$, if $\square(\aleph_{n+1})$ holds, then there exists an $\aleph_{n+1}$-Souslin tree.

mu-complete Suslin trees on mu^+

We prove that mu = mu^{< mu}, 2^mu = mu^+ and ``there is a non reflecting stationary subset of mu^+ composed of ordinals of cofinality < mu'' imply that there is a mu-complete Souslin tree on mu^+ .

树的消失层

The vanishing levels of a tree

本文研究正规 $κ$-树消失层集合的谱 $Vspec(κ)$，证明其不变性与封闭性，并给出 Aronszajn、Souslin 树在多种相容模型中的消失层可能性。

Aronszajn trees, square principles, and stationary reflection

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(κ)$ introduced by Brodsky and Rinot for the purpose of constructing $κ$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $κ$ but the stronger is not. We then prove that, if $μ$ is a singular cardinal, $\square_μ$ implies the existence of a special $μ^+$-tree with a $\mathrm{cf}(μ)$-ascent path, thus answering a question of Lücke.

一个所有 $C$-序列均为平凡的新模型

A new model for all $C$-sequences are trivial

本文构造了一个新模型，使所有 $C$-序列均为平凡，同时存在具有完全消失层的 $κ$-Souslin 树，从而回答了 Lambie-Hanson 和 Rinot 的问题，并实现了紧致性与非紧致性的最优组合。

树的威力

The power of trees

本文研究了树的有限幂在拓扑性质上的差异，提出两种构造：一种是aleph_1树其区间拓扑完美正常但平方不计数元紧凑；另一种是 inaccessible κ 下的树，其n导出树为Souslin，(n+1)导出树为特殊。

带后继函数的完全n分支序数树的初等理论

The elementary theory of full $n$-branching ordinal trees with successor functions

该论文研究带后继函数的完全n分支序数树的初等理论，对标准树按初等等价分类，建立量词消去与可判定性结果，还发展可实现性理论并澄清树与一元二阶逻辑的关系。

Full normalization for mouse pairs

We develop the theory of meta-iteration trees, that is, iteration trees whose base "model" is itself an ordinary iteration tree. We prove a comparison theorem for meta-iteration strategies parallel to the one for ordinary iteration strategies, and use it to show that the iteration strategy component of a mouse pair condenses to itself under weak tree embeddings. These constitute a class of embeddings between iteration trees that is significantly larger than the class of embeddings mentioned in the definition of mouse pair. We then use this very strong hull condensation property of mouse pairs …

Rooted trees, strong cofinality and ample generics

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.