A New Construction of Non-Extendable Intersecting Families of Sets

Abstract

In 1975, Lovász conjectured that any maximal intersecting family of $k$-sets has at most $\lfloor(e-1)k!\rfloor$ blocks, where $e$ is the base of the natural logarithm. This conjecture was disproved in 1996 by Frankl and his co-authors. In this short note, we reprove the result of Frankl et al. using a vastly simplified construction of maximal intersecting families with many blocks. This construction yields a maximal intersecting family $\mathbb{G}_{k}$ of $k-$sets whose number of blocks is asymptotic to $e^{2}(\frac{k}{2})^{k-1}$ as $k\rightarrow\infty$.