On Maximal Left-Compressed Intersecting Families Generated by a Collection of Subsets of [$n$]

We provide a characterization of maximal left-compressed families based on their generating sets $\mathcal{G}\subseteq 2^{[n]}$. We show that there is a one-to-one correspondence between maximal left-compressed families $\mathcal{A}\subseteq \binom{[n]}{k}$ and principal generating sets. Moreover, we give a complete description of maximal left-compressed intersecting families having exactly two maximal generators. Based on this, for any $X\subseteq [2,n]$, we compare $\mathcal{A}(X)$, where $\mathcal{A}$ is a rank-2 maximal left-compressed intersecting family with exactly two maximal generators, and $\mathcal{S}(X)$, where $\mathcal{S}$ denotes the Star family.