On the global linear Zarankiewicz problem

Abstract

The `global' Zarankiewicz problem for hypergraphs asks for an upper bound on the number of edges of a finite $r$-hypergraph $V$ in terms of the number $|V|$ of its vertices, assuming the edge relation is induced by a fixed $K_{k, \dots, k}$-free $r$-hypergraph $E$, for some $k\in\mathbb N$. In [4], such bounds of size $O(|V|^{r-1})$ were achieved for a semilinear $E$, namely, definable in a linear o-minimal structure. We establish the same bounds in five new settings: when $E$ is definable in (a) a semibounded o-minimal structure and the vertex set of $V$ is `sufficiently distant', (b) a model of Presburger arithmetic, (c) the expansion $\langle\mathbb R,<,+, \mathbb Z\rangle$ of the real ordered group by the set of integers, (d) a stable 1-based structure without the finite cover property, and (e) a locally modular regular type in a stable theory, such as the generic type of the solution set of the Heat differential equation. Our methods include techniques for reducing Zarankiewicz's problem to the setting of arbitrary subgroups of powers of groups, used in geometric cases (a)--(c). They also include an abstract version of Zarankiewicz's problem for general saturated `linear structures' that yields the desired bounds in the model-theoretic settings (d)--(e), as well as a parametric version in (b). Furthermore, the bounds in (a) characterise those o-minimal structures that do not recover a global field, and in (c) they yield new versions of Zarankiewicz's problem for certain ordered abelian groups.