  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.
