Thomas Jech, Saharon Shelah
Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite character.
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Partitioning the real line into Borel sets
本文研究实数线可被划分为多少个 Borel 集或闭集，证明除 Hausdorff 的 $\aleph_1$ 结果外，可能基数的谱在力迫扩张中可相当任意。
Shelah's own proof to his recent polarized partition theorem involving a singular strong limit that violates the GCH is presented. The proof is slightly re-arranges so that no use of the ideal I[λ] is made. The proof should be readabel to any student with knowledge of set theory.
The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph_1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpinski a hundred years ago.
Computable aspects of the Laver partition theorem
该文研究Laver分划定理的反数学与Weihrauch度，明确其证明论强度上下界及相关问题的算术Weihrauch度，为其可计算方面提供精确刻画。
Comment to "Almost disjoint sets, the dense set problem and the partition calculus"
本文指出1976年Baumgartner论文中引理6.10的证明存在问题，并提出修正，同时证明了一个更强的结果，其证明方法与原文相似。
A paper on ordinal partitions by Erdős and Milner (1972) has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo-Fraenkel set theory. The work is part of a project on formalising the partition calculus. The chosen material is particularly appropriate in view of the substantial corrections later published by its authors, illustrating the potential value of formal verification.
Cross-constraint basis theorems and products of partitions
本文综述并扩展了陆刘的技术，证明了n元组三色Ramsey定理不能可计算归约到n元组两色Ramsey定理的有限乘积。
If phi is a scattered order type, mu a cardinal, then there exists a scattered order type psi such that psi->[phi]^1_{mu,aleph_0} holds.
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
We show that large sets in Ellentuck topology (i.e. sets which are not nowhere Ramsey) do not admit Kuratowski's partition. The similar result is true for the Sacks real forcing.
Partition genericity and pigeonhole basis theorems
本文提出分区通用性概念，证明余超免疫集与 Kurtz 随机集均具有该性质，并由此回答 Kjos-Hanssen 和 Liu 的问题，统一推出多个鸽巢基定理。
In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $λ$, a strong coloring on $λ^+$ by $λ$ colors over a partition can be stretched to one with $λ^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds…
Partition theorems for expanded trees
本文研究扩展树的大子树划分定理，通过保持层次相等性而非高度保持，得到不依赖大基数的一致性结果，并应用于模型论问题。
Extending a result of K. Milliken \cite{Mi2}, in this paper we prove a Ramsey classification result for equivalence relations defined on uniform families of finite strong subtrees of a finite sequence $(U_i)_{i\in d}$ of fixed trees $U_i$, $i\in d$, that have a finite uniform branching but are of infinite length.
Partitions of Baire space into compact sets
本文在CH下构造了贝尔空间到紧集的不可摧毁分割，回答了Newelski的问题，并证明Shelah超幂模型满足a=a_T，从而在可测基数下得到ℵ₁<d<a=a_T的一致性。
It is proved that no non-meager subspace of the space $[ω]^ω$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.
We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.
We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees.
The Halpern-Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern-Läuchli theorem, and the same forcing from Harrington's proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.
Partition theorems for Ketonen-Solovay largeness: Hardy-like version
本文提出了一种关于Ketonen-Solovay大小集合的划分定理，证明了在特定条件下颜色函数存在大同质子集，并给出了该界限的紧性证明。
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