On symmetric 3-wise intersecting families

Abstract:

A family of sets is said to be symmetric if its automorphism group is transitive, and $3$-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 $\{1,2,\dots ,n\}$, 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.