On Maximal Left-Compressed Intersecting Families Generated by a Collection of Subsets of [$n$]
Tuan Nguyen
TL;DR
This work studies maximal left-compressed intersecting families (MLCIF) on $k$-subsets of $[n]$ and characterizes them via generating sets. It introduces the generating framework $\mathcal{F}(\mathcal{G})$ and its left-compression closure $\mathcal{H}(\mathcal{G})$, proving a bijection: every MLCIF $\mathcal{A}$ corresponds uniquely to a principal generating set $\mathcal{G}$ with $\mathcal{A}=\mathcal{F}(\mathcal{G})\cup\mathcal{F}(\mathcal{H}(\mathcal{G}))$ (Theorem 1.3). For rank-2 MLCIFs with exactly two maximal generators, the paper gives an explicit classification $\mathcal{A}=\mathcal{F}([a,b])\cup\mathcal{F}({1}\cup[b-a+2,b])$ with $b>2a-1$, and the rank-2 specialization $\mathcal{A}=\mathcal{F}([2,b])\cup\mathcal{F}({1,b})$ with $4\le b\le k+1$, along with exact size formulas such as $|\mathcal{A}|=\binom{n-1}{k-1}-\binom{n-b}{k-1}+\binom{n-b}{k-b+1}$. The authors further provide $|\mathcal{A}(X)|$ expressions for $|X|=d$ and compare them to the Star family $\mathcal{S}(X)$, establishing when $|\mathcal{A}(X)|\le|\mathcal{S}(X)|$ for large $n$. These results yield a complete, generating-set-based description of rank-2 MLCIFs and lay groundwork for higher-rank generalizations.
Abstract
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.