Club Chang's Conjecture

Global Chang's Conjecture and singular cardinals

We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{ω^ω}$.

Adjoining only the things you want: a survey of Strong Chang's Conjecture and related topics

We survey some old and new results on strong variants of Chang's Conjecture and related topics.

关于$\mathbb{P}_{max}$扩张中的$\omega$-强可测基数

On $ω$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions

本文证明了在特定Chang型决定性模型的$\mathbb{P}_{max}$扩张中，特定基数上的club filter限制到HOD上构成超滤，从而回答了Ben-Neria和Hayut提出的问题4.11。

迈向Chang模型的泛型绝对性定理

Towards a generic absoluteness theorem for Chang models

本文引入新方法建立含所有普遍贝尔实数集的模型的泛型绝对性结果，基于超紧基数等假设证明相关导出模型性质，还得出可定义幂集等结构的理论无法被力迫改变的结论。

Projective Determinacy from long Chang's Conjecture

Consider the property $(\aleph_{ω+ 1},\aleph_{ω+ 2},\ldots) \twoheadrightarrow (\aleph_1,\aleph_2,\ldots)$. Here we will show that this property with the addition of the General Continuum Hypothesis implies projective determinacy. Of particular interest here is the use of a variant covering argument to prove limited instances of mouse reflection. We believe that this approach could find use for other forms of Chang's Conjecture as well.

Approximating diamond principles on products at an inaccessible cardinal

We isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik's overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal $θ$ without changing cofinalities or adding fast clubs to $θ$. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined wi…

Chang's Conjecture with $\square_{ω_1, 2}$ from an $ω_1$-Erdős Cardinal

Answering a question of Sakai, we show that the existence of an $ω_1$-Erdős cardinal suffices to obtain the consistency of Chang's Conjecture with $\square_{ω_1, 2}$. By a result of Donder this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of $\square_{λ, 2}$ and $(λ^+, λ) \twoheadrightarrow (κ^+, κ)$ for uncountable $κ$.

Generalizations of Martin's Axiom, weak square, weak Chang's Conjecture, and a forcing axiom failure

We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{ω_1, ω_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.

A variant of Shelah's characterization of Strong Chang's Conjecture

Shelah considered a certain version of Strong Chang's Conjecture, which we denote $\text{SCC}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted $\text{SCC}^{\text{split}}$, and prove an analogous characterization of it. In particular, $\text{SCC}^{\text{split}}$ is equivalent to the assertion that the the Friedman-Krueger poset is semiproper. This strengthens and sharpens the results of Cox, and sheds some light on problems from Usuba and Torres-Perez and Wu.

Covering with Chang models over derived models

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

Chang的猜想可能在超紧基数上失败（提交）

Chang's conjecture may fail at supercompact cardinals (submitted)

我们证明了一个修订版本的拉弗的不灭性定理略有改进的经典结果。当$κ$是超紧的时，应用程序产生$(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$的一致性。实际证明表明，$ω\_1$ -回归Kurepa-树在超紧基数上是一致的，即使${\rm MM}$在所有正则基数上破坏了它们。这个相当矛盾的事实与一般的直觉相矛盾。

来自决定性模型的共尾强Chang猜想

The Cofinal Strong Chang Conjecture from Models of Determinacy

通过修改Woodin的证明，表明从决定性模型使用P_max力迫得到的模型满足更强的共尾版本的强Chang猜想，并由此证明Namba力迫是半正常的且Θ^{UB}=ω_3的一致性，以及Woodin (*)_{UB}公理蕴含该猜想。

Chang猜想、弱反射原理与$ω_2$上的树性质

Chang's Conjecture, The Weak Reflection Principle and the Tree Property at $ω_2$

我们证明了张氏猜想的一个强版本，等价于$ω_2$上的弱反射原理，连同$2^ω=ω_2$，意味着不存在$ω_2$ -Aronszajn树.

On the consistency of local and global versions of Chang's Conjecture

We show that for many pairs of infinite cardinals $κ> μ^+ > μ$, $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ for every successor cardinal $κ$ and every $μ< κ$, answering a question of Foreman.