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A note on Steinhorn's omitting types theorem

Akito Tsuboi (2009)

Colloquium Mathematicae

Let p(x) be a nonprincipal type. We give a sufficient condition for a model M to have a proper elementary extension omitting p(x). As a corollary, we obtain a generalization of Steinhorn's omitting types theorem to the supersimple case.

Amenability and Ramsey theory

Justin Tatch Moore (2013)

Fundamenta Mathematicae

The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey-theoretic reformulation of amenability constitutes a considerable weakening of the Følner criterion. As a by-product, it will be shown that in any non-amenable group G, there is a subset E of G such that no finitely additive probability measure on G measures all translates of E equally. The analysis of discrete groups will be generalized to the setting...

Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker (2014)

Fundamenta Mathematicae

In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering G L ( V ) , where V is the countably infinite-dimensional...

Automorphisms of models of bounded arithmetic

Ali Enayat (2006)

Fundamenta Mathematicae

We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) = I f i x ( j ) for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here I f i x ( j ) is the...

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and κ ω , , 2 . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then μ ( T , α ) = 2 . (3) If α < ω₁ and T ⊢ V ≠ OD, then μ ( T , α ) 0 , 2 . (4)...

Dynamical properties of the automorphism groups of the random poset and random distributive lattice

Alexander S. Kechris, Miodrag Sokić (2012)

Fundamenta Mathematicae

A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property....

Free actions of free groups on countable structures and property (T)

David M. Evans, Todor Tsankov (2016)

Fundamenta Mathematicae

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

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