SPEAKER:  Peter Keevash (Caltech)

TITLE:  Set systems with restricted intersections

ABSTRACT:  A fundamental problem of combinatorics is to determine the
maximum size of a set system, given certain constraints on the size of the
the intersection of any two sets in the system. This question initially
arose in statistics regarding the design of experiments, and has found
applications to a number of areas of mathematics and theoretical computer
science. In this talk I will discuss two important results in this area,
and a couple of applications. One is a theorem of Frankl and Wilson, which
can be used to construct explicit Ramsey graphs: these are closely related
to the problem of extracting randomness. The second is a theorem of Frankl
and Rodl, which has been used to give lower bounds for communication
complexity. I will also describe a theorem that Benny Sudakov and I
recently proved about cross-intersecting systems, and a related conjecture
of Ahlswede, Cai and Zhang.