Remarks on Schur's conjecture

Let P be a set of n>d points in Rd for d≥2. It was conjectured by Zvi 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 P⊂R3 of n points which satisfy the condition that the lengths of their sides belong to the set of k largest distances determined by P