A New Construction of Non-Extendable Intersecting Families of Sets

In 1975, Lovász conjectured that any maximal intersecting family of k-sets has at most b(e− 1)k!c 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 Gk of k-sets whose number of blocks is asymptotic to e(k2 ) k−1 as k →∞.