## An “ordered Ramsey” theorem

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $I$ be an open interval. Ordered Ramsey Theorem (Peterzil and Starchenko) Let $S_1, \dots, S_k \subseteq M^2$ be definable, and assume that $I^2 \subseteq S_1 \cup \cdots \cup S_k$. Then there exist $l \in \{1, \dots, k\}$ and an open…

## The Monotonicity Theorem

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. Let $f:I \longrightarrow M$ be definable, with $I = (a,b)$ an interval in $M$. Definition We call $f$ strictly monotone if $f$ is either constant, or strictly increasing, or strictly decreasing. Our first goal is to prove the Monotonicity Theorem There are…

## O-minimal structures

Let ${\cal M}$ be an expansion of a dense linear order $(M,\lt)$. We call ${\cal M}$ o-minimal if every definable subset of $M$ is a finite union of points and intervals. Examples (without details) By quantifier elimination, every dense linear order without endpoints is o-minimal. Let ${\cal V} = (V,\lt,+,(\lambda_k)_{k \in K})$ be an ordered…

## Expansions of dense linear orders

(For some details and references, see Chapter 1 of my notes.) We fix a structure \${\cal M} = (M,