Short proof of two cases of Chv\'atal's conjecture

TL;DR

This paper provides concise, self-contained proofs for two special cases of Chvátal's conjecture, confirming the conjecture when the family is within two stars or has rank at most three.

Contribution

The paper offers simplified proofs for two previously established cases of Chvátal's conjecture, enhancing understanding and accessibility.

Findings

Confirmed the conjecture for families within two stars

Validated the conjecture for families with rank at most three

Provided shorter, self-contained proofs

Abstract

In 1974 Chv\'atal conjectured that no intersecting family $\mathcal{F}$ in a downset can be larger than the largest star. In the same year Kleitman and Magnanti proved the conjecture when $\mathcal{F}$ is contained in the union of two stars, and Sterboul when $\operatorname{rank}(\mathcal{F})\le 3$. We give short self-contained proofs of these two statements.