Model Theory Seminar

Every archimedean real closed field is rigid, i.e., has no nontrivial automorphisms. What happens in the non-archimedean case? Shelah showed it is consistent that there are uncountable rigid non-archimedean real closed fields. Enayat asked what happens in the countable case. I believe the question is even interesting in the finite transcendence degree case. In this talk I will describe Shelah's proof and discuss some interesting phenomenon that arises even in transcendence degree 2.

I will present a transfer principle in structural Ramsey theory from finite structures to ultraproducts. In joint work with Bartosova, Dzamonja and Scow, we show that under certain mild conditions and assuming CH, when a class of finite structures has finite small Ramsey degrees, the ultraproduct has finite big Ramsey degrees for internal colorings. All Ramsey-theoretic definitions will be provided, and if time permits, I will give a sketch of the proof.

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Among the classical properties of unstable theories defined by Shelah, our understanding of the strict order hierarchy, $\text{NSOP}_n$, has remained relatively limited past $n = 4$ at the greatest. Methods originating from stability theory have given insight into the structure of stronger unstable classes, including simple and $\text{NSOP}_1$ theories. In particular, syntactic information about formulas in a first-order theory often corresponds to semantic information about independence in a theory's models, which generalizes phenomena such as linear independence in vector spaces and algebraic independence in algebraically closed fields. We discuss how the fine structure of this independence reveals exponential behavior within the strict order hierarchy, particularly at the levels $\text{SOP}_{2^{n+1}+1}$ for positive integers $n$. Our results suggest a potential theory of independence for $\text{NSOP}_n$ theories, for arbitrarily large values of $n$.

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In combinatorics, the Spencer-Shelah random graph sequence is a variation on the independent-edge random graph model. We fix an irrational number $a \in (0,1)$, and we probabilistically generate the n-th Spencer-Shelah graph (with parameter $a$) by taking $n$ vertices, and for every pair of distinct vertices, deciding whether they are connected with a biased coin flip, with success probability $n^{-a}$. On the other hand, in model theory, an $R$-mac is a class of finite structures, where the cardinalities of definable subsets are particularly well-behaved. In this talk, we will introduce the notion of 'probabalistic $R$-mac' and present an incomplete proof that the Spencer Shelah random graph sequence is an example of one.

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I will discuss work on burden 2 or dp-rank 2 expansions of theories of densely ordered Abelian groups. Such theories allow for some variety in the topological properties of definable subsets in their models and I'll discuss how diverse the collection of definable subsets in a model may be. For example, is it possible to simultaneously define an infinite discrete set and a dense co-dense subset? Answer to such questions often hinge on whether one is working in the inp-rank or dp-rank case (i.e. whether one assumes NIP or not). I will provide definitions in the talk of all the relevant notions. This is joint work with John Goodrick.

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I call a subset of the domain of a countable model absolutely undefinable if the set of its images under automorphisms of the model is uncountable. By the Kueker-Reyes theorem, all sets that are not absolutely undefinable are parametrically definable in $L_{\omega_1 \omega}$. I will survey classical results about first-order undefinability in the standard model of arithmetic, and I will contrast them with some old and some new results about absolute undefinability in nonstandard models.

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This talk will discuss my work with Miriam Parnes and four undergraduates which took place last summer at an REU at Towson University. We say that a class of structures in some fixed language is indivisible if, for all structures A in the class and number of colors k, there is a structure B in the class such that, no matter how we color B with k colors, there is a monochromatic copy of A in B. Parnes and I became interested in this property when studying the classification of theories via positive combinatorial configurations. In this talk, following the work with our students, I will examine indivisibility on classes of graphs. In particular, we will look at hereditarily sparse graphs, cographs, perfect graphs, threshold graphs, and a few other classes. This work is joint with Felix Nusbaum, Zain Padamsee, Miriam Parnes, Christian Pippin, and Ava Zinman.

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This talk is about a hybrid of finite and infinite model theory, where formulas can reference two kinds of non-logical symbols. There are 'built-in' or 'interpreted' symbols, which are interpreted over an infinite structure for a complete background theory - e.g. real or integer arithmetic. And there are 'uninterpreted relation symbols' ranging over finite relations living within an infinite model of the theory. The expressiveness of first order logic in this setting was studied in the late 90's and early 2000's, motivated by the real field case, stemming from questions in databases and computational geometry. Much of this talk will review what was accomplished decades ago. We then present an overview of recent results by myself and Ehud Hrushovski (https://arxiv.org/abs/2304.09231), that begin to prorgess on the many questions that were left open in decades past.

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