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⊨T . Let ϕ(x;d) ∈ L($\mathcal{U}$) a formula, non-forking over M . Then there is θ(y) ∈ tp(d/M) such that the partial type {ϕ(x;d′): d′ ∈ θ($\mathcal{U}$)} is consistent.
'Invariant types in NIP theories' (2015)
It seems Itay Kaplan has recently solved this conjecture with 'A definable (p,q) -theorem for NIP theories'(2022).
I believe this proof can potentially be generalized to the case where only ϕ(x,y) has NIP with some minor restrictions since Itay's proof only relies implicitly on T having NIP in a couple of places. Does anyone know of any counterexamples to the definable (P,Q)-theorem in classes outside NIP (e.g. in NTP2)?
As far as I know the conjecture is still open for NIP formulas outside NIP theories. I solved it for the case of having VC-codensity less than $2$ (with $q=2$) here [1]. I will be uploading this week the newest version of the document to ArXiV, after the referee comments, which make it way more readable, so you might want to wait before taking a look. Nevertheless, you might be interested in the questions at the very end.
[1] Definable (ω,2)-theorem for families with VC-codensity less than 2, Pablo Andújar Guerrero, https://arxiv.org/abs/2210.04551