The global Zarankiewicz's problem for hypergraphs asks for an upper bound on the number of edges of a hypergraph, whose edge relation is induced by a fixed hypergraph \(E\) that has no sub-hypergraphs of a given size. Basit-Chernikov-Starchenko-Tao-Tran (2021) obtained linear Zarankiewicz bounds in the case of a semilinear \(E\), namely \(E\) definable in a linear o-minimal structure. Moreover, those bounds characterised linearity among all o-minimal structures.
In this talk, we extend this theorem to a broad range of "linear-like" structures, in o-minimal, Presburger Arithmetic and stability theoretic settings. Among others, we characterise combinatorially those o-minimal structures that do not expand a real closed field.
Joint work with Aris Papadopoul