arXiv subjects

Nicholas Ramsey

Publications and source records attributed to Nicholas Ramsey.

23 records · Page 2Linked to original sources

arXiv2019

On Kim-Independence

We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

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arXiv2018

A note on NSOP$_{1}$ in one variable

We prove that, in order to establish that a theory is NSOP$_{1}$, it suffices to show that no formula in a single free variable has SOP$_{1}$.

math.LO↗

arXiv2018

Generic expansion and Skolemization in NSOP$_1$ theories

We study expansions of NSOP$_1$ theories that preserve NSOP$_1$. We prove that if $T$ is a model complete NSOP$_1$ theory eliminating the quantifier $\exists^{\infty}$, then the generic expansion of $T$ by arbitrary constant, function, and relation symbols is still NSOP$_1$. We give a detailed analysis of the special case of the theory of the generic $L$-structure, the model companion of the empty theory in an arbitrary language $L$. Under the same hypotheses, we show that $T$ may be generically expanded to an NSOP$_1$ theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP$_1$ theories, adding instances of algebraic independence to their conclusions.

math.LO↗

arXiv2018

Local character of Kim-independence

We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.

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arXiv2016

On model-theoretic tree properties

We study model theoretic tree properties ($\text{TP}, \text{TP}_1, \text{TP}_2$) and their associated cardinal invariants ($κ_{\text{cdt}}, κ_{\text{sct}}, κ_{\text{inp}}$, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem ($\text{TP} \Rightarrow \text{TP}_1 \lor \text{TP}_2$) for countable theories, show that $\text{TP}_1$ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak $k-\text{TP}_1$ is equivalent to $\text{TP}_1$ (answering a question of Kim and Kim). Besides, we give a characterization of $\text{NSOP}_1$ via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed $\text{NSOP}_1$.

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