## 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.