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

TL;DR

This paper proves Chvátal's conjecture for all downsets of the Boolean lattice where each subset has size at most three, extending understanding of intersecting systems in finite set families.

Contribution

It establishes the validity of Chvátal's conjecture for small-rank downsets, specifically those with subsets of size up to three.

Findings

Chvátal's conjecture holds for all downsets with subsets of size at most three.

The proof extends the class of downsets for which the conjecture is verified.

Provides insights into the structure of intersecting families in small-rank downsets.

Abstract

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.