Independent Sets in Algebraic Hypergraphs

TL;DR

This paper explores the structure of algebraic hypergraphs in algebraically closed fields, demonstrating that dense hypergraphs contain dense low-dimensional algebraic subsets, and introduces a generalized fiber dimension theorem.

Contribution

It adapts the hypergraph containers method to algebraic hypergraphs and generalizes the classical fiber dimension theorem in algebraic geometry.

Findings

Dense algebraic hypergraphs contain dense low-dimensional algebraic subsets.

A generalized fiber dimension theorem for algebraic geometry is established.

The method bridges combinatorics and algebraic geometry in a novel way.

Abstract

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