An upper bound on the number of Steiner triple systems. Issue 4 (5th March 2013) - Linial, Nathan; Luria, Zur
Abstract Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n ‐vertex Steiner triple systems: <![CDATA[articlemathrsfs, amsmath, amssymbemptydocumentalign* STS(n) = ( (1+o(1))ne^2 )^n^26align*document]]> <inline-graphic xlink:href="ark:/27927/pgg3qnzxng0" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. Our main result is that <alternatives> <![CDATA[articlemathrsfs, amsmath, amssymbemptydocumentalign* STS(n) ≤ ( (1+o(1))ne^2 )^n^26. align*document]]> </alternatives> 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‐Lovász theorem it can be shown that <alternatives> <![CDATA[articlemathrsfs, amsmath, amssymbemptydocumentalign* F(n) ≤ ( (1+o(1))ne^2 )^n^22. align*document]]> </alternatives> We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 399–406, 2013