Minimum number of affine simplices of given dimension

Abstract

In this paper we formulate and solve extremal problems in the Euclidean space Rd 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 R3, then S contains at least n4−cn3 affine simplices for some constant c. A function related to Sperner’s Theorem and its well-known extension to reciprocal sums is also considered and its relation to Turán’s hypergraph problems is discussed.