Saharon Shelah
Let P be a distinguished unary predicate and K= {M: M a model of cardinality aleph_n with P^M of cardinality aleph_0}. We prove that consistently for n=4, for some countable first order theory T we have: T has no model in K whereas every finite subset of T has a model in K. We then show how we prove it also for n=2, too.
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On compactness of weak square at singulars of uncountable cofinality
该研究探讨了在不可数余数的奇异点处弱方框原理的紧致性，并展示了在某些假设下其紧致性及更强假设的不足。
Stationary Reflection and the Failure OF SCH at $\aleph_{ω_1}$
针对Ben-Neria、Hayut和Unger的问题，本文以不可数多个超紧致基数存在为假设，证明了ℵ_{ω₁}是强极限且2^ℵ_{ω₁}>ℵ_{ω₁+1}、ℵ_{ω₁+1}的每个驻集都反射的一致性，实现了驻集反射与SCH在ℵ_{ω₁}处失效的结合。
Good Scales and Non-Compactness of Squares
本文研究了正方形原理在奇异基数上的紧性问题，证明了在特定条件下，ℵω上的所有尺度都是良好的，但正方形原理*在ℵω上不成立，同时某些内部可接近性失败。
We prove that for any regular kappa and mu > kappa below the first fix point (lambda = aleph_lambda) above kappa, there is a graph with chromatic number > kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic number less than or equal to kappa.
We prove a compactness theorem for pseudopower operations of the form $pp_{Γ(μ,σ)}(μ)$ where $\aleph_0<σ=cf(σ)\leq cf(μ)$. Our main tool is a result that has Shelah's cov vs. pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set $A$ of regular cardinals for which $pcf(A)$ has an inaccessible accumulation point.
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 model for global compactness
本文在超可compact基数一致性的假设下，构造了一个集合论模型，其中每个奇异基数λ上存在一个均匀超滤子，满足对于所有比λ的紧致度大的基数θ，该超滤子是θ-不可分解的，并展示了色数紧致性的实例，推翻Hedetniemi猜想的反例在ZFC中的最优性。
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
该研究围绕集合论全域V与HOD的紧性现象，证明了奇异强极限基数的幂集分歧特性，回答了Hayut的问题，并揭示Magidor覆盖定理的最优性。
It is known that the assumption that ``GCH first fails at \aleph_ω'' leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that this is not so in ZF: namely there is a generic cardinal-preserving extension of L (or any universe of ZFC + GCH in which all ZF axioms hold, the axiom of choice fails, GCH holds for all cardinals \aleph_n, but there is a surjection from PowerSet(\aleph_ω) onto λ, where λ is any previously chosen cardinal in L greater than \aleph_ω, for instance, \aleph_{ω+17}. In other words, in such an extension GCH holds in proper sense for all cardinals \aleph_n but …
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
Shelah's conjecture fails for higher cardinalities
本文构造编码 $\aleph_{α+1}$-Kurepa 树的 $\mathcal{L}_{ω_1,ω}$-语句，证明在更高基数版本中局部 Hanf 数可不存在，并给出非绝对的合流谱。
We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0}. We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced separable abelian p-group in lambda. (Note: both results fail if lambda = lambda^{aleph_0} or if lambda is strong limit, cf (mu)= aleph_0< mu).
We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal an…
We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$ when $κ$ is supercompact. The actual proofs show that $ω\_1$-regressive Kurepa-trees are consistent above a supercompact cardinal even though ${\rm MM}$ destroys them on all regular cardinals. This rather paradoxical fact contradicts the common intuition.
The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $κ$ is singular strong limit, then $2^κ=κ^+$. We prove that given a singular cardinal $κ$ of {\em cofinality} $η$ in the ground model, which is a limit of suitable large cardinals, and $η^+=\aleph_γ$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $η$, such that $κ$ becomes $\aleph_{γ+η}$, and SCH fails at $κ$. Furthermore, if $η$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_η$. Our large c…
We prove that colouring of pairs from aleph_2 with strong properties exists. The easiest to state (and quite a well known problem) it solves: there are two topological spaces with cellularity aleph_1 whose product has cellularity aleph_2 ; equivalently we can speak on cellularity of Boolean algebras or on Boolean algebras satisfying the aleph_2-c.c. whose product fails the aleph_2-c.c. We also deal more with guessing of clubs.
Echeloned saturation and forcing axioms
针对Paul Larson的问题，研究结合不同力迫公理与集合论假设，证明特定条件下ℵ₁上无弱Laver理想，以及Baumgartner公理下ℵ₂上无Laver理想。
In this paper we prove that if k is a cardinal in L[0^#], then there is an inner model M such that M |= (V_k,E) has no elementary end extension. In particular if 0^# exists then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than aleph_1 of uncountable cofinality in L[0^#] is Mahlo in every strict inner model of L[0^#].
Failure of singular compactness for Hom
在$V=L$假设下，构造了一个$χ$-自由阿贝尔群，其在Hom函子下表现出奇异紧致性的失效，为非平凡对偶性提供了相容的反例。
We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω_1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.
Saturation of reduced products
本文研究可数结构约化积的饱和性，证明稳定性决定其饱和程度及与超幂同构的条件，并将结论推广到 $F_\sigma$、分层及解析 P-理想。
Telgársky's conjecture states that for each $k \in \mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\mathsf{GCH}+\square$, that if {\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $π$-base with certain nice properties, then {\s…
Tight $σ$-filtrations and the BFPP of $C(ω^*)$
在P(ω)紧σ-滤过的假设下，证明C(ω*)不满足球不动点性质，利用Cc(ω₁)与P(ω)紧σ-滤过性的等价性，还得到相关布尔代数商的性质。
Quadruples and cubes
本文在ZFC中证明阶梯立方体与四元组关系的失败条件，并利用奇妙理想建立后继基数上的正极化关系，同时否定连续基数上的此类理想存在。
Strong Completeness of Provability Logic for Uncountable Languages
本文利用Erdős-Rado定理证明了对于特定基数的模态语言，GL逻辑在广义Icard拓扑下不满足强完备性，并引入了两种拓扑空间类以证明GL和GL.3在特定语言下满足强完备性。
We prove that the class of all ordinals Ord is not weakly compact with respect to definable classes. Specifically, in any model of ZFC, the definable tree property fails for Ord, in that there is a definable Ord tree with no definable cofinal branch; the definable partition property fails, in that there is a definable 2-coloring of pairs from a certain definable proper class, with no definable homogeneous proper class; and the definable compactness property fails for $\mathcal{L}_{\infty,ω}$, in that there is a definable theory in this logic all of whose set-sized subtheories are satisfiable, …
Weak Baumgartner axioms and universal spaces
本文研究完美波兰空间上鲍姆加特纳公理的两个弱化形式，证明其中一个仍保留原公理的许多重要推论，另一个虽独立于ZFC但在Cohen和随机模型中失败，并引入回避类空间对来区分这些弱公理。
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