Peter’s Solution to Exercise 1

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 $A_m\neq\emptyset$.…

Sparse subsets of the plane

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…

O-minimality and uniform finiteness

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal? Lemma The following are equivalent: every ${\cal N} \equiv {\cal M}$ is o-minimal; for every…

Finite fibers of constant size

Let ${\cal M}$ be an expansion of a dense linear order \$(M,