Given a countable, totally ordered commutative monoid $\mathcal{R}=(R,\oplus,\leq,0)$, with least element $0$, there is a countable, universal and ultrahomogeneous metric space $\mathcal{U}_\mathcal{R}$ with distances in $\mathcal{R}$. We refer to this space as the $\mathcal{R}$-Urysohn space, and consider the theory of $\mathcal{U}_\mathcal{R}$ in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free $n^{\text{th}}$ roots of the complete graph (e.g. the random graph when $n=2$), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of $\text{Th}(\mathcal{U}_\mathcal{R})$ by algebraic properties of $\mathcal{R}$, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOP$_n$-hierarchy. Using the submonoid of idempotents in $\mathcal{R}$, we also characterize superstability, supersimplicity, and weak elimination of imaginaries. Finally, we give necessary conditions for elimination of hyperimaginaries, which further develops previous work of Casanovas and Wagner.
We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.
We show that there are uncountably many countable lattices. We give a discussion of which such lattices can be modular or distributive. The method applies to show that certain other classes of structures also have uncountably many homogeneous structures
We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we answer questions related to our research.
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.
We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fra\" {i}ss\' e limits) to study closed amenable subgroups $G$ of the symmetric group $S_\infty$ of a countable set, where $S_\infty$ has the topology of pointwise convergence. We construct $G$-invariant measures on the universal minimal flows associated with these groups $G$ in, moreover, an algorithmic manner. This leads to an identification of the generic elements, in the sense of being Martin-L\" of random, of these flows with respect to the constructed invariant measures. Along…
The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klavžar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed…
We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below $\aleph_ω$. We introduce $\aleph_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.
Einstein Institute of Mathematics, The Hebrew University of Jerusalem（希伯来大学爱因斯坦数学研究所）；Department of Mathematics, Rutgers University（罗格斯大学数学系）
We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of $\aleph_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.
Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. Here we consider a class of countable metrically homogeneous graphs. The algebra of an age is a concept introduced by Cameron and is closely connected to the profile of the automorphism group of the associated countable structure. Cameron later provided sufficient structural conditions on the age of $\aleph_0$-categorical countable homogeneous structures for showing that the algebra of the age is a polynomial algebra. In this paper, we use Cameron's result to deduce th…
We show that, for a coanalytic subspace $X$ of $2^ω$, the countable dense homogeneity of $X^ω$ is equivalent to $X$ being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^ω$ such that $X^ω$ is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a…
We prove that the Fraïssé limit of a Fraïssé class $\mathcal C$ is the (unique) countable structure whose isomorphism type is comeager (with respect to a certain logic topology) in the Baire space of all structures whose age is contained in $\mathcal C$ and which are defined on a fixed countable universe. In particular, the set of groups isomorphic to Hall's universal group is comeager in the space of all countable locally finite groups and the set of fields isomorphic to the algebraic closure of $\mathbb F_p$ is comeager in the space of countable fields of characteristic $p$.
We show that for any countable homogeneous ordered graph $G$, the conjugacy problem for automorphisms of $G$ is Borel complete. In fact we establish that each such $G$ satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of $G$ is Borel reducible to the conjugacy relation on automorphisms of $G$.
We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that ever…
In 1959, P. Hall introduced the locally finite group $\mathcal{U}$, today known as Hall's universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in $\mathcal{U}$. It can be explicitly described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraisse limit of the class of all finite groups. Since its introduction Hall's group, and several natural generalisations, have been widely studied. In this article we use a generalisation of Fraisse theory to construct a countable, univ…
We construct a homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid.
It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. Our construction over the 2-element field is related to the Reed--Muller codes.
Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces. We show that an indivisible metric space must be totally Cantor disconnected, which implies in particular that every Urysohn space U_V with V bounded or not but dense in some initial segment of R+, is divisible. On the other hand we also show that one can remove "large" …
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；University of Wrocław（弗罗茨瓦夫大学）；University of Münster（明斯特大学）；Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院）
We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $M^{eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $D$ being a reduct of a binary random structure. Somewhat more preciely: (a) if for every $n \geq 2$, …
The random ordered graph is the up to isomorphism unique countable homogeneous linearly ordered graph that embeds all finite linearly ordered graphs. We determine the reducts of the random ordered graph up to first-order interdefinability.
Let $\mathcal{R}=(R,\oplus,\leq,0)$ be an algebraic structure, where $\oplus$ is a commutative binary operation with identity $0$, and $\leq$ is a translation-invariant total order with least element $0$. Given a distinguished subset $S\subseteq R$, we define the natural notion of a "generalized" $\mathcal{R}$-metric space, with distances in $S$. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of $S$. We first construct an ordered additive structure $\mathcal{S}^*$ on the space of quantifier-free $2$-types cons…
We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_\infty$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q}, <)$.
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.
Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained …
Transformation monoids carry a canonical topology --- the topology of point-wise convergence. A closed transformation monoid $\mathfrak{M}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{K}$ of structures, if every monoid-isomorphism of $\mathfrak{M}$ to the endomorphism monoid of a member of $\mathcal{K}$ is automatically a homeomorphism. In this paper we show automatic homeomorphicity-properties for the monoid of non-decreasing functions on the rationals, the monoid of non-expansive functions on the Urysohn space and the endomorphism-monoid of the countable unive…
Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well-known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $ω_1$ which has the same extension property as $R$, yet its group of automorphisms is trvial. We also present a similar, although technically more complicated, construction of a complete metric space of density $ω_1$, having the extension property like t…
