Minimum number of affine simplices of given dimension

Abstract

In this paper we formulate and solve extremal problems in the Euclidean space R d and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge between the original problems and the presented extremal theorem on set systems. As a sample corollary, it follows that if no triple is collinear in a set S of n points in R 3 , then S contains at least n 4 − cn 3 affine simplexes for some constant c. A function related to Sperner's theorem and the YBLM inequality is also considered and its relation to hypergraph Turán problems is discussed.