关于极化划分关系的等价性

Every CBER is smooth below the Carlson-Simpson generic partition

Let $E$ be a countable Borel equivalence relation on the space $\mathcal{E}_{\infty}$ of all infinite partitions of the natural numbers. We show that $E$ coincides with equality below a Carlson-Simpson generic element of $\mathcal{E}_{\infty}$. In contrast, we show that there is a hypersmooth equivalence relation on $\mathcal{E}_{\infty}$ which is Borel bireducible with $E_1$ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on $ω$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $ω$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) i…

Shelah最近分拆定理的一个证明

A proof of Shelah's recent partition theorem

谢拉自己的证明，他最近的极化分割定理涉及一个奇异的强极限，违反了GCH。这个证明稍微重新安排了一下，这样就不使用理想I[λ]了。任何有集合论知识的学生都可以读到这个证明。

基数在（λ，2 ^lambda）中的配分关系的相容性

Consistency of partition relation for cardinal in (lambda,2^lambda)

我们提出了一个使2 ^lambda爆炸的强迫，并使"许多正极化配分关系“（在我们的主要定理（c）中精确的意义上）在区间[lambda，2 ^lambda]中保持。这推广了[276]第1节的结果，并且强迫是那里强迫的“多基数”版本。

A characterization of ordinal analysis

Ordinal analysis induces a partition of $Σ^1_1$-definable and $Π^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $Π^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $Σ^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.

密度之下的诸多划分关系

Many partition relations below density

该研究通过力迫法构造大基数情形下的极化划分关系，将其应用于一般拓扑学问题，拓展了集合论与拓扑学的交叉研究。

划分演算与基数不变量

Partition Calculus and Cardinal Invariants

证明了对区间$[\aleph_1,\aleph]$中的每个基数同时应用$θ$在$ω$之上的强极化关系是一致的。我们的结论是，这种积极的关系是一致的，每一个基数不变的连续。我们发现，类似的结果举行超紧基数，并在一定的假设下，一个强极限奇异。

有限强一致子树族的一个典范划分定理

A canonical partition theorem for uniform families of finite strong subtrees

推广了K. Milliken \cite{Mi2}中，本文证明了定义在固定树有限序列$(U_i)_{i\in d}$的有限强子树的一致族上的等价关系的Ramsey分类结果，其中固定树序列$U_i$，$i\in d$具有有限一致分支但具有无限长.

不稳定类的Karp复杂性

The Karp complexity of unstable classes

如果对于所有基数λ，L_{infty,lambda} -等价的关系将K划分为等价类的集合（而不是真类），则结构类K是受控的。我们证明了双传递线性序类是受控的，而任何具有ω-独立性质的伪初等类都是不受控的。