Extending partial isometries of generalized metric spaces

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.Comment: 12 pages, final version incorporating referee comment

Forking and dividing in Henson graphs

For $n\geq 3$, define $T_n$ to be the theory of the generic $K_n$-free graph, where $K_n$ is the complete graph on $n$ vertices. We prove a graph theoretic characterization of dividing in $T_n$, and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, $T_n$ provides a counterexample to a recent question of Chernikov and Kaplan.Comment: 11 page

A remark on strict independence relations

We prove that if $T$ is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for $T$ and strict independence relations for $T^{\text{eq}}$. We use this observation to show that if $T$ is the theory of the Fra\"{i}ss\'{e} limit of finite metric spaces with integer distances, then $T^{\text{eq}}$ has more than one strict independence relation. This answers a question of Adler [1, Question 1.7].Comment: 9 pages, to appear in Archive for Mathematical Logi

Distance structures for generalized metric spaces

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 consistent with the axioms of $\mathcal{R}$-metric spaces with distances in $S$, and show that, if $A$ is an $\mathcal{R}$-metric space with distances in $S$, then any model of $\text{Th}(A)$ logically inherits a canonical $\mathcal{S}^*$-metric. Our primary application of this framework concerns countable, universal, and homogeneous metric spaces, obtained as generalizations of the rational Urysohn space. We adapt previous work of Delhomm\'{e}, Laflamme, Pouzet, and Sauer to fully characterize the existence of such spaces. We then fix a countable totally ordered commutative monoid $\mathcal{R}$, with least element $0$, and consider $\mathcal{U}_\mathcal{R}$, the countable Urysohn space over $\mathcal{R}$. We show that quantifier elimination for $\text{Th}(\mathcal{U}_\mathcal{R})$ is characterized by continuity of addition in $\mathcal{R}^*$, which can be expressed as a first-order sentence of $\mathcal{R}$ in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.Comment: 30 page

Stability in a group

We develop local stable group theory directly from topological dynamics, and extend the main results in this subject to the setting of stability "in a model". Specifically, given a group $G$, we analyze the structure of sets $A\subseteq G$ such that the bipartite relation $xy\in A$ omits infinite half-graphs. Our proofs rely on the characterization of stability via Grothendieck's "double-limit" theorem (as shown by Ben Yaacov), and the work of Ellis and Nerurkar on weakly almost periodic $G$-flows.Comment: 26 pages, final version incorporating referee comments (substantial revision from previous version

Neostability in countable homogeneous metric spaces

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.Comment: 32 page

Model theoretic properties of the Urysohn sphere

We characterize model theoretic properties of the Urysohn sphere as a metric structure in continuous logic. In particular, our first main result shows that the theory of the Urysohn sphere is $\text{SOP}_n$ for all $n\geq 3$, but does not have the fully finite strong order property. Our second main result is a geometric characterization of dividing independence in the theory of the Urysohn sphere. We further show that this characterization satisfies the extension axiom, and so forking and dividing are the same for complete types. Our results require continuous analogs of several tools and notions in classification theory. While many of these results are undoubtedly known to researchers in the field, they have not previously appeared in publication. Therefore, we include a full exposition of these results for general continuous theories.Comment: 23 pages, some proofs shortened, appendix adde

Pseudofinite groups and VC-dimension

We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta(x;\bar{y})$, we prove various aspects of "local fsg" for the right-stratified formula $\delta^r(x;\bar{y},u):=\delta(x\cdot u;\bar{y})$. This includes a $\delta^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on $\delta^r$-formulas, and generic compact domination for $\delta^r$-definable sets.Comment: 18 page

Independence in generic incidence structures

We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n = 2$, this is the theory of existentially closed projective planes. We give an $\forall\exists$-axiomatization of $T_{m,n}$, show that $T_{m,n}$ does not have a countable saturated model when $m,n\geq 2$, and show that the existence of a prime model for $T_{2,2}$ is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for $T_{m,n}$. We show that $T_{m,n}$ is NSOP$_1$, but not simple when $m,n\geq 2$, and we show that $T_{m,n}$ has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence

Stable groups and expansions of (Z,+,0)(\mathbb{Z},+,0)

We show that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacin and Sklinos, we conclude that $(\mathbb{Z},+,0)$ has no proper stable expansions of finite weight. A corollary of this result is that if $P\subseteq\mathbb{Z}^n$ is definable in a finite dp-rank expansion of $(\mathbb{Z},+,0)$, and $(\mathbb{Z},+,0,P)$ is stable, then $P$ is definable in $(\mathbb{Z},+,0)$. In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.Comment: 10 pages, final version to appear in Fundamenta Mathematica