Independent sets in algebraic hypergraphs

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, the spirit of so-called transference principles extremal combinatorics, that if a given algebraic hypergraph “dense” certain sense, then generic low-dimensional subset its vertices induces subhypergraph also “dense.” (For technical reasons, only consider subsets are parameterized by rational functions.) proof approach inspired containers method, developed Balogh, Morris, and Samotij independently Saxton Thomason (although adapting method setting presents some unique challenges do not occur when working with finite hypergraphs). Along way, establish natural generalization classical dimension fibers theorem geometry, which interesting own right.