Pierre Simon conjectured a model-theoretic definable version of Matoušek’s $(p, q)$-theorem in NIP theories:
[Conjecture 5.1]: Let $T$ be NIP and $M \models T$. Let $\phi(x;d)\in L(\mathcal{U})$ a formula, non-forking over $M$. Then there is $\theta(y)\in tp(d/M)$ such that the partial type {$\phi(x;d'):d'\in \theta(\mathcal{U})$} is consistent.
'Invariant types in NIP theories' (2015)
This problem has been worked on over the years and solved for particular cases such as dp-minimal and distal theories. It appears that Itay Kaplan has recently solved this conjecture with 'A definable $(p, q)$-theorem for NIP theories'(2022).
I'm curious if anyone has any ideas for interesting examples, applications and, more particularly, extensions to this problem as I'm looking to explore this further.
You might want to take a look at the open questions I list at the very end of this preprint [1]. I will upload probably today or tomorrow the newest version to ArXiv, after the referee comments, which has more readable proofs. The questions at the very end do not change however, except that I note that Kaplan has given a positive answer to question (2) in NIP theories.
I actually emailed Noga Alon at some point asking whether the $(p,q)$ problem for VC classes had been considered as far as he knows for $p$ or both $p$ and $q$ infinite cardinals and he said no.
It is my impression that the definable (p,q)-theorem likely has unexplored applications in NIP model theory, for example (very vagely):
In connection to Kaplan, Bays and Simon's recent study of compressible types.
Topological, in characterizing a usable notion of NIP definable compactness.
In studying uniform definability of types and strict pro-definability of the space of definable types.
About point (3), I gave a presentation where I went a bit in depth about it. I could send you the slides if you'd like.
[1] Definable (ω,2)-theorem for families with VC-codensity less than 2, Pablo Andújar Guerrero, https://arxiv.org/abs/2205.13665