Combinatorics Seminar

When: Sunday, May 20, 10am Where: Schreiber 309 Speaker: Zur Luria, Hebrew University Title: Upper bounds on the number of Steiner triple systems and 1-factorizations

Abstract:

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 show 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 the entropy method, the basic idea of which is to estimate the cardinality of a set X by sampling uniformly from that set and then estimating the entropy of this process using tools from the field of information entropy.

Joint work with Nati Linial.