On symmetric 3-wise intersecting families

David Ellis, Bhargav Narayanan

Published 2016-08-10Version 1

A family of sets is said to be {\em symmetric} if its automorphism group is transitive, and {\em $3$-wise intersecting} if any three sets in the family have nonempty intersection. Frankl conjectured in 1981 that if $\mathcal{A}$ is a symmetric $3$-wise intersecting family of subsets of $\{1,2,\ldots,n\}$, then $|\mathcal{A}| = o(2^n)$. Here, we give a short proof of Frankl's conjecture, using a `sharp threshold' result of Friedgut and Kalai.