Flag Algebras and Exact Turán Densities for Geometric Hypergraphs

This monograph establishes exact Turán densities for a class of geometric hypergraphs using the framework of Flag Algebras introduced by Razborov. We focus on r-uniform geometric hypergraphs Hn defined on point sets in general position in Rd, where hyperedges are induced by specific geometric constraints related to convexity and order types. By encoding the geometric properties into the semantic structure of the flag algebra calculus, we derive a sequence of semidefinite programming (SDP) relaxations that provide rigorous upper bounds on the extremal densities. Specifically, we resolve the Turán density for the forbidden geometric configuration of a "crossing 3-cycle" in the plane, proving it to be exactly 2/9, a result previously conjectured in the context of geometric Ramsey theory. Furthermore, we extend the differential method to the geometric setting, characterizing the stability of the extremal constructions. The results demonstrate the efficacy of automated theorem proving techniques in continuous combinatorial geometry.