A. S. Kechris, V. G. Pestov, S. Todorcevic
We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We formulate a property strengthening the Disjoint Amalgamation Property and prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Furthermore, we prove indivisibility for every Fraisse structure in an arbitrary finite relational language satisfying this property. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse…
A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl …
We consider Fraïssé structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure, a single structure which codes the big Ramsey degrees of a given Fraïssé structure. This in turn leads to the definition of a completion flow; we show that if a Fraïssé structure admits a big Ramsey structure, then the automorphism group admits a unique universal completion flow. We also discuss the problem of w…
Fraïssé limit and Ramsey Theorem: the case of MV-algebras and a categorical generalization
该研究刻画了有限MV-代数的Fraïssé极限并证明其满足Ramsey性质，还将结论推广至一般范畴，得出满足Kechris-Pestov-Todorcevic对应最低条件时，有限结构范畴与其完备子范畴的Ramsey性质等价的结论。
We develop the theory of Fraïssé limits for classes of finite-dimensional multi-seminormed spaces, which are defined to be vector spaces equipped with a finite sequence of seminorms. We define a notion of a Fraïssé Fréchet space and we use the Fraïssé correspondence in this setting to obtain many examples of such spaces. This allows us to give a Fraïssé-theoretic construction of $(\mathbb{G}^ω, (\|\cdot\|_n)_{n<ω})$, the separable Fréchet space of almost universal disposition for the class of all finite-dimensional Fréchet spaces with an infinite sequence of seminorms. We then identify and pro…
Roelcke non-precompactness, non-simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are examined among others.
Moore characterized the amenability of automorphism groups of countable ultrahomogeneous structures by a Ramsey-type property. We extend this result to automorphism groups of metric Fraïssé structures, which encompass all Polish groups. As an application, we prove that amenability is a G_delta condition.
Fraïssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition
本文研究了允许有限单形分解的结构的有限大Ramsey度，并证明了此类结构的有限大Ramsey度当且仅当其最小单形分解中的每个单形部分也具有有限大Ramsey度。该结果依赖于对单形结构的有限大Ramsey度的特征化，即可数单形结构具有有限大Ramsey度当且仅当可通过具有有限大Ramsey度的链进行链化。有趣的是，这两种特征化都需要对链的深层结构性质的理解。Fraïssé猜想（实际上是由Laver解决的正向结果）在特征化具有有限大Ramsey度的单形结构中起关键作用，而分析允许有限单形分解的结构的大Ramsey组合学则需要一个关于大Ramsey度的乘积Ramsey定理。我们发现这一结果特别引人注目，因为大Ramsey度在一般乘积陈述中表现不佳。在附录中，我们结合Oudrar和Pouzet最近的结果以及我们对允许有限单形分解的结构的有限大Ramsey度的分析，来特征化所有可数关系结构中，其语言具有线性序且年龄具有多项式增长时的有限大Ramsey度的存在性。
We prove that the automorphism group of a Fraïssé structure M equipped with a notion of stationary independence is universal for the class of automorphism groups of substructures of M. Furthermore, we show that this applies to certain homogeneous n-gons.
We present three examples of countable homogeneous structures (also called Fraisse limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures. Our first example is a particular case of a rather general construction on Fraisse classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraisse classes, the mixed sums, leading to a Fraisse class with all finite symmetric groups appearing as…
In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fraïssé structures and apply it to prove the non-amenability of the automorphism groups of the directed graph $\mathbf{S}(3)$ and the Boron tree structure $\mathbf{T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $\mathrm{GL}(\mathbf{V}_\infty)$, where $\mathbf{V}_\infty$ is the countably infinite dimensional vector space over a finite field $F_q$, we show that the unique invariant measure on the univer…
We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the randomly ordered rational Urysohn space.
In this paper we investigate the connections between Ramsey properties of Fraisse classes K and the universal minimal flow M(G_K) of the automorphism group G_K of their Fraisse limits. As an extension of a result of Kechris, Pestov and Todorcevic we show that if the class K has finite Ramsey degree for embeddings, then this degree equals the size of M(G_K). We give a partial answer to a question of Angel, Kechris and Lyons showing that if K is a relational Ramsey class and G_K is amenable, then M(G_K) admits a unique invariant Borel probability measure that is concentrated on a unique generic …
In this paper we introduce a new topological Ramsey space whose elements are infinite ordered polyhedra. Then, we show as an application that the set of finite polyhedra satisfies two types of Ramsey property: one, when viewed as a category over $\mathbb N$; the other, when considered as a class of finite structures. The (ordered) random polyhedron is the Fraisse limit of the class of finite ordered polyhedra; we prove that its group of automorphisms is extremely amenable. Finally, we present a countably infinite family of topological Ramsey subspaces; each one determines a class of finite ord…
Under no additional assumptions, in this paper we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation and recent results about weak Fraïssé categories. Namely, generalizing the fact that every Ramsey class has amalgamation, we show that classes with finite Ramsey degrees have weak amalgamation. We then invoke the machinery of weak Fraïssé categories to perform the construction. This improves previous similar results where an analogous construction was ca…
We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraisse classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraisse limit of any Fraisse class that is the class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sen…
We investigate correspondences between extreme amenability and amenability of automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type properties of their smooth classes, similar to Kechris, Pestov and Todorcevic, and Tatch Moore. In particular, we focus on some Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions. Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed. Moreover, we pr…
Exact big Ramsey degrees for finitely constrained binary free amalgamation classes
本文刻画有限二元语言中由有限禁用不可约子结构定义的自由合并类的精确大 Ramsey 度，并证明其 Fraïssé 极限具有强大 Ramsey 结构及可度量化通用完备流。
We show that if a subgroup of the automorphism group of the Fraisse limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite set.
We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S(G)$, the Samuel compactification, and $E(M(G))$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg's problem for several automorphism groups and give a detailed study in the case of $G = S_\infty$, leading us to define and investigate several new types of ultrafilter on a countable set.
We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(ω_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(ω)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.
This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions im…
On some Fraisse limits with free amalgamation
该研究分两部分展开，先探讨稳定Kim-forking的概念与背景，再给出通过满足强条件的Fraïssé类极限构建NSOP1理论的通用方法，明确其独立性性质，基于前人成果推进了模型论相关研究。
Universal minimal flows of homeomorphism groups of continua
通过定义射影Fraïssé范畴的近似Ramsey性质，证明了该性质与群极端可安性等价，并应用于伪螺线管同胚群得到非可度量化泛最小流。
We show that if g is a generic (in the sense of Baire category) isometry of a generic subspace of the Urysohn metric space U, then g does not extend to a full isometry of U. The same holds for the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: for a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f in Aut(M). From our analysis, a dichotomy arises and some structural results are derived that, in particular, …
Large orbits, projective Fraisse limits, and the pseudo-arc
本文研究射影 Fraïssé 极限 $\mathbb{P}$ 的自同构群在其对合上的共轭作用，证明该作用存在余稀疏轨道。
Homeomorphisms of continua through projective Fraïssé limits
本文通过投影Fraïssé极限研究伪弧的同胚群，证明了Homeo(P)对Homeo(P)^\mathbb{N}的对角共轭作用具有稠密轨道，并展示了存在一个P的同胚不与Aut(\mathbb{P})中的元素共轭。
Group-extensive embeddings into Fraïssé structures and stationary weak independence relations
本文研究弗赖谢结构的平稳弱独立关系（SWIR）与其ω-年龄群可扩张性的关系，证明具SWIR的线性序弗赖谢结构具该可扩张性，给出仅具单性质的示例，并考察含几乎所有可数无限超齐次有向图的实例的对应性质。
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories of indiscernibles" to be special kinds of companionable theories of finite structures, and much of the work in our arguments is carried in the context of the model-companion. Among other things, this approach allows us to prove that the companion of a theory of indiscernibles whose "base" consists of the quantifier-free formulas is necessarily the theory of the Fraisse limit of a Fraisse c…
Infinite-dimensional Ramsey theory for binary free amalgamation classes
本文为有限二元语言中的自由合并类Fraïssé极限发展无限维拉姆齐理论，引入A.3(2)-理想概念，在更弱假设下证明Todorcevic抽象拉姆齐定理，并恢复精确大拉姆齐度。
Using the natural action of $S_\infty$ we show that a countable hereditary class $\mathcal C$ of finitely generated structures has the joint embedding property (JEP) and the weak amalgamation property (WAP) if and only if there is a structure $M$ whose isomorphism type is comeager in the space of all countable, infinitely generated structures with age in $\mathcal C$. In this case, $M$ is the weak Fraïssé limit of $\mathcal C$. This applies in particular to countable structures with generic automorphisms and recovers a result by Kechris and Rosendal [Proc. Lond. Math. Soc., 2007].
We develop a theory of \emph{Katetov functors} which provide a uniform way of constructing Fraisse limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraisse limits.
We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an $ω$-categorical sparse graph has no $ω$-categorical expansion with extremely amenable automorphism group.
Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties
本文研究 Cantor 空间上测度自同构群中存在稠密或余瘦共轭类的判据，利用 Fraïssé 理论将好测度刻画为超齐次测度的子类，并确定了取有理值且自同构群承认余瘦共轭类的好测度。
We survey recent developments concerning two properties of classes of finite structures: the Ramsey property and the extension property for partial automorphisms (EPPA).
In this paper we prove that the automorphism groups of certain countable generic structures are not amenable. For doing that, we first prove the existence of particular matrices that do not satisfy the convex Ramsey condition. For a pair of elements in a smooth class, we introduce the property of forming a free-pseudoplane in the generic structure. We then prove the non-amenability of the automorphism group of a generic structure obtained from a smooth class with a pair that forms a free-pseudoplane. As an application we show that the automorphism group of an ab-initio generic structure that i…
This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial property which resembles the statement of Ramsey's classic theorem. Such classes of structures have been called Ramsey classes. Nesetril and Roedl showed that they have the amalgamation property, and therefore each such class has a homogeneous Fraisse-limit. Ramsey classes have recently attracted attention due to a surprising link with the notion of extreme amenab…
We define a projective Fraissé family whose limit approximates the universal Knaster continuum. The family is such that the group $\textrm{Aut}(\mathbb{K})$ of automorphisms of the Fraissé limit is a dense subgroup of the group, $\textrm{Homeo}(K)$, of homeomorphisms of the universal Knaster continuum. We prove that both $\textrm{Aut}(\mathbb{K})$ and $\textrm{Homeo}(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extreme…
We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric s…
An unusual example of a universal automorphism group
本文研究了 Fraïssé 结构的 universal 自动同构群与 group-extensible ω-age 的关系，通过构造一个例子证明两者不等价。
Counting big Ramsey degrees of the homogeneous and universal $K_4$-free graph
本文针对通用齐次无 $K_4$ 图这一特例，重新简洁呈现已有大 Ramsey 度刻画，并计算若干小图的实际大 Ramsey 度。
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results s…
Twenty years of Nešetřil's classification programme of Ramsey classes
本文回顾了Nešetřil提出的Ramsey类分类计划在过去二十年的进展，列出了未解决的问题，并讨论了向EPPA和大Ramsey结构等新领域的扩展。
Fixed points on null and tame flows for groups of automorphisms
基于 Kechris-Pestov-Todorčević 对应的推广，本文研究了 Fraïssé 结构自同构群的零流与驯流，证明了在自由联合嵌入类与自由融合类条件下相应流的不动点定理。
We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of fin…
Merges of Smooth Classes and Their Properties
本文研究了两个具有泛极限的 Fraïssé-like 类能否合并为一个具有泛极限的类，探讨了合并类的性质及其与结构 Ramsey 理论和 Hrushovski 性质（EPPA）的联系。
Transitive Extensions of Automorphism Groups of Generic Structures
研究传递扩展的存在性，发现仅当颜色数为2的幂时，边缘着色的k超图有传递扩展；k超 tournaments 在Cherlin意义上仅当k为偶数时存在传递扩展。
Embeddings into the generic poset
本文证明每个可数无限偏序集都能以余有限像方式嵌入泛型偏序集，且像的自同构可唯一扩展为整个泛型偏序集的自同构。
Cohen-like first order structures
本研究用力迫替代归纳法构造不可数泛型结构，证明其多为刚性，并给出自同构群为整数群的稠密实数集例子。
Let L be a countable elementary language, N be a Fraisse limit. We consider free amalgamation for L-structures where L is arbitrary. If free amalgamation for finitely generated substructures exits in N, then it is a stationary independece relation in the sense of K.Tent and M.Ziegler [TZ12b]. Therefore Aut(N) is universal for Aut(M) for all substructures M of N. This follows by a result of I.Müller [Mue13] We show that c-nilpotent graded Lie algebras over a finite field and c-nilpotent groups of exponent p (c < p) with extra predicates for a central Lazard series provide examples. We replace t…
The random permutation is the Fraïssé limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all structures which have a first-order definition in the random permutation.
We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and c…
The amalgamation property and Urysohn structures in continuous logic
本文研究了连续逻辑中 amalgamation 性质与 Urysohn 结构的关系，证明了相关类别的 Fraïssé 性质和 EPPA，以及自同构群的 universal 性质。
We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.
We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simp…
In 2005, the paper "Fraissé limits, Ramsey theory, and topological dynamics of automorphism groups" [KPT] by Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow. This immediately led to an explicit representation of this invariant in many concrete cases. However, in some particular situations, the framework of [KPT] does not allow to perform the computation directly, but only after a slight modification of the original argument. The purpose of the present paper is to supplement [KPT] in order to avoid that tw…
Generic dc-automorphisms of two-sorted ultrametric spaces
研究双排序超度量空间的dc-自同构，证明自同构群具有剩余共轭类，并给出部分自同构的共尾合并性质。
A footnote to the KPT theorem in structural Ramsey theory
本文通过另一种证明方式，展示了结构Ramsey理论中KPT定理的结论，即Fraïssé极限具有全序可约结构或Ramsey性质失效的条件。
Ramsey expansions of $Λ$-ultrametric spaces
本文研究了有限分布格Λ的超度量空间的Ramsey扩张，揭示了其自同构群的universal minimal flow，并证明了所有已知有限维排列结构的Ramsey性质。
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures. In 2005, Nešetřil initiated a systematic study of the so-called Ramsey classes of finite structures. This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs. A key ingradient is an explicit combinatorial algorithm to fill-in…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。