For a permutation π in the symmetric group Sn​ let the {\it total
degree} be its valency in the Hasse diagram of the strong Bruhat order on
Sn​, and let the {\it down degree} be the number of permutations which are
covered by π in the strong Bruhat order. The maxima of the total degree and
the down degree and their values at a random permutation are computed. Proofs
involve variants of a classical theorem of Tur\'an from extremal graph theory.Comment: 14 pages, minor corrections; to appear in S\'em. Lothar. Combi