On generating sets of left-compressed intersecting families
Tuan Nguyen, Thi Nguyen, Thu Tran
TL;DR
This paper characterizes when left-compressed, $k$-uniform families generated by sets $\mathcal{G}$, namely $\mathcal{F}(n,k,\mathcal{G})$, are intersecting. The central concept is strong-intersection of the generators, defined by $G\sim_{si}H$ and equivalent to the existence of $l$ with $\mu_G(l)+\mu_H(l)>l$, which yields a practical criterion for $\mathcal{F}(\mathcal{G})$ to be intersecting. The authors then show an extension principle: if $\mathcal{F}(\mathcal{G})$ is LCIF, then replacing each $G$ with $\pi(G)$ to form $\pi(\mathcal{G})$ gives $\mathcal{F}(\pi(\mathcal{G})) \supseteq \mathcal{F}(\mathcal{G})$ and preserves the left-compressed-intersecting structure. They relate their condition to Bond's theorem and discuss maximal left-compressed intersecting families (MLCIFs), posing open questions about necessary and sufficient conditions for $\mathcal{F}(\mathcal{G})$ to be MLCIF and how to extend LCIFs to MLCIFs.
Abstract
We construct a left-compressed family F(n, k, G) generated by a collection of generating sets G, and identify conditions on G under which F(n, k, G) is an intersecting family. From this, we obtain a convenient method to construct left-compressed intersecting families. After that, we provide some comparisons between the theorem of Bond and our result. Finally, we present some results on the generators of a maximal left-compressed intersecting family.