A very fundamental result of Shelah in neostability theory is the fact that any unstable NIP theory has an instance of the strict order property, a formula $\varphi(x,y)$ (with $x$ and $y$ possibly tuples of variables) and a sequence $(b_i)_{i<\omega}$ of parameters such that the for any $i<j$, $\varphi(x,b_i)$ is a strict subset of $\varphi(x,b_j)$. (This is equivalent to the theory having a definable partial order with infinite chains on some sort.)
If you follow the proof carefully you get a more local statement. First recall that a sequence $(b_i)_{i \in I}$ (for $I$ an infinite linearly ordered set) witnesses the order property with a formula $\varphi(x,y)$ if for every $i \in I$, the partial type $\{\varphi(x,b_k) : k\leq i\}\cup\{\neg \varphi(x,b_k) : k > i\}$ is consistent.
For any formula $\varphi$, we'll write $\varphi^0$ for $\varphi$ and $\varphi^1$ for $\neg \varphi$. The more careful statement you can get is this:
Fact: Fix an NIP theory $T$. Let $\varphi(x,y)$ be a formula and $(b_r)_{r \in \mathbb{Q}}$ an indiscernible sequence such that $(\varphi(x,b_r))_{r \in \mathbb{Q}}$ witnesses the order property. There is an interval $I \subseteq \mathbb{Q}$, a finite set $Q \subset \mathbb{Q}$ disjoint from $I$, and a function $f : Q \to \{0,1\}$ such that the the sequence of definable sets $(\chi(x,b_r))_{r \in I}$ is strictly nested, where $$\chi(x,y) = \varphi(x,y) \wedge \bigwedge_{r \in Q} \varphi^{f(r)}(x,b_r).$$
Really we only need the formula $\varphi(x,y)$ to be NIP.
The question I am wondering about is how big $Q$ needs to be (and moreover how big it needs to be on either side of $I$).
Question 1: For every $n$, are there examples of formulas $\varphi(x,y)$ and indiscernible sequences $(b_i)_{i\in \mathbb{Q}}$ witnessing the order property in unstable NIP theories such that any $Q$ guaranteed in the fact has size at least $n$?
Question 2: For every $n$, are there such formulas and sequences such that for any $Q$ and $I$ as in the fact, the sets $\{r \in Q: r < I\}$ and $\{r \in Q : r > I\}$ both have size at least $n$?
I'm fairly certain that at least Question 1 has a positive answer, but I'm having difficulty finding examples. You get an easy bound in terms of the alternation number of the formula, but, at least in the easy case I was able to check, this bound is not saturated. Specifically, if $\varphi(x,\bar{y})$ is a formula with $x$ a single variable in $\mathsf{DLO}$, then for any indiscernible sequence $(\bar{b}_i)_{i\in \mathbb{Q}}$, if $(\varphi(x,\bar{b}_i))_{i \in \mathbb{Q}}$ is a witness to the order property, then I believe that either it is already strictly nested or the formula $\varphi(x,\bar{b}_r)\wedge \varphi(x,\bar{b}_0)$ is strictly nested for $r \in (0,\infty)$. (The same should be true in any o-minimal theory.) This is despite the fact that there are such $\varphi(x,\bar{y})$ with arbitrarily high alternation numbers.
It feels like this might be related to dp-minimality, but I can't see how to get larger $Q$'s in "$\mathsf{DLO}^n$" either. That said, I haven't actually ruled those out as examples yet either.
