The global Zarankiewicz's problem can be stated in terms of intersections of relations with grids as follows: Let \(E\) be an \(r\)-ary relation. Suppose \(E\) contains no grids of a given size. Then the intersection number with any grid grows slowly. (A grid is a cartesian product of \(r\) many finite sets.) We sketch the proof of this theorem in case \(E\) is definable in a model of Presburger Arithmetic, and in the expansion \((\mathbb{R}, <, +, \mathbb{Z})\) of the real ordered group by the integers. In particular, we obtain a paramateric version of the corresponding theorem for \((\mathbb{R}, <, +)\) where the parameters can now also vary over any Presburger set.
Joint work with Aris Papadopoulos