Patrick Speissegger
Assume $\mathcal M$ and $S$ satisfy all of the assumptions in the exercise, including 1-4. For each $m\in\mathbb N$, define $A_m = \{x\in\Pi_n(S) : |S_x| \geq m\}$. We will show that each $A_m$ is either empty or equal to $S$, and hence that $S_x$ is equal to the minimal value of $m$ such that $A_m\neq\emptyset$.…
Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $S \subseteq M^2$. Our goal is to show that if $S$ is definable and sparse, then $S$ satisfies uniform finitess. For $z \in S$, we say that $\Pi_1\rest{S}$ is a homeomorphism at $z$ if there exists an open box $B \subseteq…
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 expansion of a dense linear order $(M,