On symmetric 3-wise intersecting families

TL;DR

This paper proves Frankl's conjecture that symmetric 3-wise intersecting families of subsets are asymptotically negligible compared to the power set size, using a sharp threshold result.

Contribution

The paper provides a concise proof of Frankl's conjecture leveraging Friedgut and Kalai's sharp threshold theorem.

Findings

Confirmed that symmetric 3-wise intersecting families are o(2^n) in size.

Applied sharp threshold techniques to combinatorial set families.

Simplified the proof of a longstanding conjecture.

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.