Upper bounds on the number of Steiner triple systems and 1-factorizations

יום א', 04/03/2012 - 14:00

A 1-factorization of the complete graph Kn is a partition of its edges into n-1 perfect matchings. A Steiner triple system on [n] = {1,...,n} is a collection T of triples such that each pair in [n] is contained in a unique triple.

We will discuss the connections between these (and other) objects, and present previously known bounds on their number. We'll prove that the number of 1-factorizations of Kn is at most ((1+o(1)) n/e^2)^(n^2/2) and that the number of Steiner triple systems on [n] is at most ((1+o(1)) n/e^2)^(n^2/6).

The proofs make use of information entropy.

Joint work with Nati Linial.