An upper bound on the number of Steiner triple systems

Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n-vertex Steiner triple systems: STS(n) = ( (1 + o(1)) n e2 )n2 6 . Our main result is that STS(n) ≤ ( (1 + o(1)) n e2 )n2 6 . The proof is based on the entropy method. As a prelude to this proof we consider the number F (n) of 1factorizations of the complete graph on n vertices. Using the KahnLovász theorem it can be shown that F (n) ≤ ( (1 + o(1)) n e2 )n2 2 . We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp.