Closed Intersecting Families of finite sets and their applications

Paul Erd\H{o}s and L\'aszl\'o Lov\'asz established that any \emph{maximal intersecting family of $k-$sets} has at most $k^{k}$ blocks. They introduced the problem of finding the maximum possible number of blocks in such a family. They also showed that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. Later P\'eter Frankl, Katsuhiro Ota and Norihide Tokushige used a remarkable construction to prove the existence of a maximal intersecting family of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks. In this article we introduce the notion of a \emph{closed intersecting family of $k-$sets} and show that such a family can always be embedded in a maximal intersecting family of $k-$sets. Using this result we present two examples which disprove two special cases of one of the conjectures of Frankl et al. This article also provides comparatively simpler construction of maximal intersecting families of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks.