Chv\'atal's conjecture for downsets of small rank

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,…,n}, denoted by 2^{[n]}, is at most 2^{n-1}, with one of the extremal structures being the family comprised of all subsets of [n] containing a fixed element, called as a star. A longstanding conjecture of Chv'atal aims to generalize this simple observation for all downsets of 2^{[n]}. In this note, we prove this conjecture for all downsets where every subset contains at most 3 elements.