Patrick Speissegger
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…
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 $M$. We start by translating the condition “$f$ is strictly increasing” into a condition involving two subsets of $I^2$: on the one hand, we have the triangle $\displaystyle \Delta(I):=…
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…