We know that, for any finite family F of intervals, if any two intersect then the whole family has nonempty intersection. More generally, if F has the (p,2)-property (for any p intervals in F, there are always 2 among them that interect), then there exist at most 2(p-1) points such that every interval in F contains at least one of them. A (p,q) theorem is a generalization of this kind of result, for different classes of sets.

We describe the mathematical development from the classical Helly theorem on convex sets to the Alon-Kleitman-Matoušek (p,q) theorem for VC (or NIP) classes. We will go over some applications, conjectures and improvements around this result in the field of model theory.