## Combinatorics Seminar

When: Sunday, December 18, 10am
Where: Schreiber 309
Speaker: David Ellis, Queen Mary University of London
Title: Some applications of the p-biased measure to
Erdős-Ko-Rado type problems
## Abstract:

If X is a finite set, the p-biased measure on the power-set of X is
defined as follows: choose a subset S of X at random by including each
element of X independently with probability p. If FF is a family of
subsets of X, one can consider the p-biased measure of FF, denoted by
\mu_p(FF), as a function of p; if FF is closed under taking supersets,
then this function is an increasing function of p. Seminal results of
Friedgut and Friedgut-Kalai give criteria for this function to have
a "sharp threshold". A careful analysis of the behaviour of this function
also yields some rather strong results in extremal combinatorics which
do not explicitly mention the p-biased measure - in particular, in
the field of Erdős-Ko-Rado type problems, which concern the sizes of
families of objects in which any two (or threeâ€¦) of the objects "agree"
or "intersect" in some way. We will discuss some of these, including
a recent proof of an old conjecture of Frankl that a symmetric three-wise
intersecting family of subsets of {1,2,...,n} has size o(2^n), and
a sharp bound on the size of the union of two intersecting families of
k-element subsets of {1,2,...,n}, for k≤n/2-o(n). Based on joint work
with (subsets of) Nathan Keller, Noam Lifshitz and Bhargav Narayanan.