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/e(2))n(2)/6. Our main result is that STS(n) <= ((1 + o(1))n/e(2))n(2)/6. The proof is based on the entropy method. As a prelude to this proof we consider the number F(n) of 1 -factorizations of the complete graph on n vertices. Using the Kahn-Lovasz theorem it can be shown that F(n) <= ((1 + o(1))n/e(2))n(2)/2. We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp. (c) 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 399-406, 2013
steiner triple systems asymptotics the entropy method combinatorial enumeration