Some applications of the p-biased measure to Erdős-Ko-Rado type problems

11 October 2016

'Erdős-Ko-Rado type problems' are well-studied in extremal combinatorics; they concern the sizes of families of objects in which any two (or any $r$) of the objects in the family 'agree', or 'intersect', in some way.

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 $\mathcal{F}$ is a family of subsets of $X$, one can consider the $p$-biased measure of $\mathcal{F}$, denoted by $\mu_p(\mathcal{F})$, as a function of $p$. If $\mathcal{F}$ is closed under taking supersets, then this function is an increasing function of $p$. Seminal results of Friedgut and Friedgut-Kalai give criteria under which this function has a 'sharp threshold'. Perhaps surprisingly, 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. 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 $\{1,2,\ldots,n\}$ has size $o(2^n)$, and some 'stability' results characterizing the structure of 'large' $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots,n\}$. Based on joint work with (subsets of) Nathan Keller, Noam Lifschitz and Bhargav Narayanan.

  • Combinatorial Theory Seminar