## Dimension

Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$ with language $\la$. Definition Let $A \subseteq M$ and $\phi(x_1, \dots, x_n)$ be an $\la(A)$-formula, and set $S:= \phi(M^n)$. We define $$\dim_A S := \sup\set{\dim(s/A):\ \N \models \phi(s), \ \N \succ \M},$$ where $\dim(s/A)$ is defined in $\N$ as in this post. Note…