Thursday 03 Dec 2015: Counting designs
Peter Keevash (Oxford) -
A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plucker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plucker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient. In 1974, Wilson conjectured an approximate formula for the number of such systems. We will outline a proof of this conjecture, and a more general estimate for the number of Steiner systems. Our main tool is the technique of Randomised Algebraic Construction, which we introduced to resolve a question of Steiner from 1853 on the existence of designs.