Combinatorics Seminar

When: Sunday, January 18, 10am Where: Schreiber 309 Speaker: Pedro Vieira, ETH Zurich Title: Almost-Fisher Families

Abstract:

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family F of subsets of {1,2,...,n} with all pairwise intersections of size \lambda can have at most n non-empty sets. One may weaken the condition by requiring that for every set in F, all but at most k of its pairwise intersections have size \lambda. We call such families (k,\lambda)-Fisher. Vu was the first to study the maximum size of such families, proving that for k=1 the largest family has 2n-2 sets, and characterizing when equality is attained. In this talk we present a refined version of these results, showing how the size of the maximum family depends on the parameter \lambda. In particular we show that for small \lambda one essentially recovers Fisher's bound. We also solve the next open case of k=2 and obtain the first non-trivial upper bound for general k.

Joint work with S. Das and B. Sudakov.