Remarks on Schur's conjecture ?
Let P  be a set of n>d n > d points in Rd R d for d?2 d ? 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?2 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 P ? R 3 of n points which satisfy the condition that the lengths of their sides belong to the set of k largest distances determined by P.
schur's conjecture regular simplices diameter graphs