On Schurʼs conjecture

Let P be a set of n points in Rd. It was conjectured by Schur that the maximum number of (d−1)-dimensional regular simplices of edge length diam(P), whose every vertex belongs to P, is n. We prove this statement under the condition that any two of the simplices share at least d−2 vertices. It is left as an open question to decide whether this condition is always satisfied. We also establish upper bounds on the number of all 2- and 3-dimensional simplices induced by a set of n points P⊂R3 which satisfy the condition that the lengths of their sides belong to the set of k largest distances determined by P.