Patrick Speissegger
First some terminology: Let $X$ be a set and $Y_1, \dots, Y_l \subseteq X$, and put $\Y:= \{Y_1, \dots, Y_l\}$. We say that the $\Y$ partitions $X$ if $X = Y_1 \cup \cdots \cup Y_l$ and the $Y_j$-s are pairwise disjoint. Given $Z \subseteq X$, we say that $\Y$ is compatible with $Z$ if, for…
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 $S \subseteq M^2$. $S$ is sparse if $S$ has empty interior. Lemma Assume $S$ is definable. The following are equivalent: $S$ is sparse; the set $S’$ of all $x \in M$ such that $S_x$ is infinite is finite; $S$ is…
Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. For $A \subseteq M$, the definable closure of $A$ is defined by $$\dcl(A):= \set{b \in M:\ \{b\} \text{ is } A\text{-definable}}.$$ Exercise Let $A \subseteq M$. Prove that $\dcl(A) = \acl(A)$. Let $\phi(x)$ be a formula with parameters in $A$. Prove that…
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…