Search Results

I am wondering about a natural generalization of theorem 1.4 in the article Dvoretzky's theorem — Thirty years later by Milman. My first thought was to look at Milman's paper that he cites for the res …

Recall that in a first-order theory $T$, a hyperimaginary is an equivalence class of some type-definable equivalence relation $E(x,y)$ (with $x$ a possibly infinite tuple of parameters). The $E$-equiv …

For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$ be the ordinary covering numbers. …

Fix a complete first order theory $T$ and a set of parameters $A$ in the monster model $\mathcal{U}$. Recall that an $A$-invariant global type is a type $p(x) \in S_x(\mathcal{U})$ which is fixed by a …

In 'Elementary pairs of models' by Bouscaren, she mentions with a remark at the end that if $T$ is a superstable theory then $T$ has a Vaughtian pair if and only if $T^\text{eq}$ has a Vaughtian pair, …

It is a well known fact that any intersection of equivalence relations on a fix set is itself an equivalence relation. The same holds for a few other common relational theories, such as partial orders …

After talking to a couple of people, I have been unable to determine whether this is even known. Recall that a first-order theory $T$ has the strict order property or SOP if there is a formula $\varph …

It is a well-known fact that while while Robinson arithmetic can interpret surprisingly strong theories, it cannot interpret $\mathsf{I}\Delta_0 + \mathrm{Exp}$, i.e., Peano arithmetic with induction …

A very fundamental result of Shelah in neostability theory is the fact that any unstable NIP theory has an instance of the strict order property, a formula $\varphi(x,y)$ (with $x$ and $y$ possibly tu …

A topological space $X$ has the fixed-point property if any continuous map $f : X \to X$ has a fixed point (i.e., an $x$ such that $f(x) = x$). It's well known that certain topological spaces, such as …

Let $(P_r)_{r \in \mathbb{R}}$ be an $\mathbb{R}$-indexed family of propositional variables. Let $\mathcal{L}$ be the collection of all propositional sentences formed from the variables $(P_r)_{r \in …

Let $X$ be a compact metrizable space and let $\mathcal{K}_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric f …

These are fairly standard terms, but for the sake of completeness: An ultrafilter $\mathcal{U}$ on $\omega$ is a p-point if whenever $(A_n)_{n<\omega}$ is a partition of $\omega$ such that $A_n \notin …

Stable theories have the following useful property, which I will state in a sub-optimal way for simplicity's sake: Fact 1. If $T$ is $\lambda$-stable for some $\lambda \geq |T|^+$, then for any set o …

Let $\mathcal{L}$ be an infinite relation language (this question is trivial in a finite relational language). Suppose that $\mathcal{K}$ is the class of finite models of some $\mathcal{L}$-theory (si …