On Kim-independence

Kaplan, Itay; Ramsey, Nicholas

doi:10.4171/jems/948

Cited by 48 publications

(118 citation statements)

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“…In this section, we analyze model theoretic properties of T m,n and we characterize various notions of independence including Kim independence, dividing independence, and forking independence. For background and details on these notions, see [4] and [16].…”

Section: Remark 22 T Cmentioning

confidence: 99%

“…1 . We will demonstrate NSOP 1 using a characterization due to Chernikov, Kaplan, and Ramsey [8,16] involving the existence of a ternary relation satisfying certain axioms (see the proof of Theorem 4.11 below). We first recall the following classical ternary relations, as well as several axioms of an arbitrary ternary relation.…”

Section: Kim Independence and Nsopmentioning

confidence: 99%

“…[16], Theorem 9.1). Suppose | is an invariant ternary relation on M which satisfies strong finite character and witnessing, the independence theorem, existence over models, monotonicity over models, and symmetry over models, then T is NSOP 1 and | = | K .…”

mentioning

confidence: 97%

“…The main tool for proving that generic theories are NSOP 1 is the independence relation criterion developed by Chernikov and Ramsey [8] and refined by Kaplan and Ramsey [16]: If there is a notion of independence between subsets of the monster model of a complete theory, which satisfies certain axioms, then that theory must be NSOP 1 , and the independence relation must be Kim independence. Applying this criterion to T m,n , we obtain a characterization of Kim independence in terms of incidence-free disjoint amalgamation of algebraically closed sets (Definition 4.5, Theorem 4.11).…”

mentioning

confidence: 99%

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Independence in Generic Incidence Structures

Conant¹,

Kruckman²

2019

J. symb. log.

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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 ∀∃-axiomatization of T m,n , show that T m,n does not have a countable saturated model when m, n ≥ 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 ≥ 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. 1The theory T m,n is also interesting from the perspective of the model theory of graphs. The class of graphs has a model companion, the unique countable model of which is known as the "random graph". This structure has a bipartite counterpart, the "random bipartite graph". When it is viewed as a structure in a language with unary predicates for the bipartition, the theory of the random bipartite graph is the model companion of the theory of arbitrary incidence structures. Both the random graph and the random bipartite graph have well-understood theories: they are ℵ 0 -categorical, simple, unstable, and have quantifier elimination and trivial algebraic closure.In the same way, T m,n can be viewed as the theory of existentially closed bipartite graphs omitting the complete bipartite graph K m,n . Thus the theories T m,n are bipartite counterparts to the theories T n of generic K n -free graphs (existentially closed graphs omitting a complete subgraph of size n). The theories T n , introduced by Henson [13], are important examples of countably categorical theories exhibiting the properties TP 2 , SOP 3 , and NSOP 4 (see [24], and also [9] for discussion of the model theoretic properties of Henson graphs). We show that for m, n ≥ 2, T m,n also has TP 2 , but it is NSOP 1 , hence tamer, in a sense, than the Henson graphs.In another contrast to the Henson graphs, T m,n is not countably categorical when m, n ≥ 2. In fact, we show that in this case, T m,n has continuum-many types over the empty set, and hence has no countable saturated model (Theorem 3.3). The question of whether T m,n has a prime model seems to be a hard combinatorial problem; in the case m = n = 2, we show that it is equivalent to a longstanding open problem in the theory of projective planes (Theorem 3.8).Finally, from the point of view of classification theory, the generic theory recipe has been a fruitful source of examples of simple theories (see [6], for example). Recently, many examples of generic theories which are not simple have been shown to be NSOP 1 (see [8], [19], [20]). The theories T m,n are further examples of this phenomen...

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Section: Remark 22 T Cmentioning

confidence: 99%

Section: Kim Independence and Nsopmentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

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Independence in Generic Incidence Structures

Conant¹,

Kruckman²

2019

J. symb. log.

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“…Although our new theories are not simple (Theorem ), there is evidence that these theories are NSOP

_{1}

. The class of NSOP

_{1}

theories has been recently analyzed in and it is shown there that these theories provide a natural generalization of simple theories and that they have many properties of simple theories, which were important for applications in the case of ACFA. Hence, knowing that our new theories are NSOP

_{1}

may result in similar applications.…”

Section: Introductionmentioning

confidence: 91%

Model theory of fields with virtually free group actions

Beyarslan

Kowalski

2018

Proc. London Math. Soc.

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For a group G, we define the notion of a G‐kernel and show that the properties of G‐kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP2.

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Generic expansion of an abelian variety by a subgroup

d’Elbée

2021

Mathematical Logic Qtrly

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On Kim-independence

Cited by 48 publications

References 0 publications

Independence in Generic Incidence Structures

Independence in Generic Incidence Structures

Model theory of fields with virtually free group actions

Generic expansion of an abelian variety by a subgroup

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