Remarks on Schur's conjecture

Let P   be a set of n>dn>d points in RdRd for d≥2d≥2. It was conjectured by Zvi Schur that the maximum number of (d−1)(d−1)-dimensional regular simplices of edge length diam(P)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−2d−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⊂R3P⊂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.