Intersecting families of sets are typically trivial

Graph Theory Seminar
Tuesday, March 30, 2021 - 3:45pm for 1 hour (actually 50 minutes)
Location For password, please email Anton Bernshteyn (bahtoh ~at~
Lina Li – University of Waterloo –
Anton Bernshteyn

A family of subsets of $[n]$ is intersecting if every pair of its members has a non-trivial intersection. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n \geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Significantly improving their results, we show that the same conclusion holds for $n \geq 2k + 100 \ln k$. Our proof uses the Sapozhenko’s graph container method and the Das-Tran removal lemma.

This is joint work with József Balogh, Ramon I. Garcia and Adam Zsolt Wagner.