Regular Simplices in Higher Dimensions

A classical problem in combinatorial geometry, posed by Erdös in 1946, asks to determine the maximum number of unit segments in a set of n points in the plane. Since then a great variety of extremal problems in finite point sets have been studied. Here, we look at generalizations of this question concerning regular simplices. Among others we answer the following question asked by Erdös: Given n points in R6, how many triangles can be equilateral triangles? For our proofs we use hypergraph Turán theory. This is joint work with Dumitrescu and Liu.