Tapani Hyttinen, Saharon Shelah
We show that if M is a stable unsuperstable homogeneous structure, then for most kappa < |M|, the number of elementary submodels of M of power kappa is 2^kappa .
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We prove a main gap theorem for e-saturated submodels of a homogeneous structure. We also study the number of e-saturated models, which are not elementarily embeddable to each other
We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.
This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as $X/M$. In the next section, we construct some examples and analyse the topological relationship between $X$ and $X/M$. In the final section, we apply $X/M$ to get novel results about Lindelöf spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.
On the downward Löwenheim-Skolem Theorem for elementary submodels
本文通过基于形式系统中替换的新模型定义，提出了一种纯代数方法，并给出了初等子模型的下向Löwenheim-Skolem定理的新证明。
Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.
In this note, we use elementary submodels to prove that a separable monotonically normal compactum can be mapped on a separable metric space via a continuous function whose fibers have cardinality at most 2.
The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams's theorem on cycle-decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.
We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
Approximation Theory and Elementary Submodels
该研究运用集合论初等子模型方法，为逼近理论中的知名定理给出新的简短证明，部分结果更强，丰富了逼近理论的证明手段。
We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field $\mathbb{R}((\mathbb{Z}))$ are definable. Similarly, all types over the quotient field of the Witt…
We study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in [T. Hyttinen, On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematicae, 156(1998): 167-182]. We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure.
Weight, net weight, and elementary submodels
本文通过初等子模型方法，在ZFC下构造了$C(ω_1)$中净权重不可数的Hausdorff空间，并证明添加Cohen实数后正则空间净权重可数，同时解决了文献[6]中的两个主要问题。
The Lattice Problem for Models of $\mathsf{PA}$
本文综述了 Peano 算术模型格问题 60 年的研究进展，详细分析了特殊情况以解释 James Schmerl 构造具有指定结构间格初等扩张的技术，并讨论了可数递归饱和模型的初等子模型格的相关结果。
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstra…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to find dividing lines, prove their consequences, prove "structure theorems, positive theorems" on those in the "low side" (in particular stable and superstable theories), and prove "non-structure, complexity theorems" on the "high side". It has started with categoricity and number of non-isomorphic models. It is probably recognized as the central part of model …
Weakly reflecting graph properties
本文研究 L. Soukup 用于不可数图的初等子模型框架，探讨将“良反映”性质假设弱化为自然出现的较弱形式，以理解相关无限图论问题。
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