On the maximum number of points in a maximal intersecting family of finite sets

Abstract

Abstract. Paul Erdős and László Lovász proved in a landmark article that, for any positive integer k, up to isomorphism there are only finitely many maximal intersecting families of k−sets (maximal k−cliques). So they posed the problem of determining or estimating the largest number N (k) of the points in such a family. They also proved by means of an example that N (k) ≥ 2k − 2 + 1 2 2k−2 k−1 . Much later, Zsolt Tuza proved that the bound is best possible up to a multiplicative constant by showing that asymptotically N (k) is at most 4 times this lower bound. In this paper we reduce the gap between the lower and upper bound by showing that asymptotically N (k) is at most 3 times the Erdős-Lovász lower bound. Conjecturally, the explicit upper bound obtained in this paper is only double the lower bound.

On the Number of Maximal Intersecting $k$-Uniform Families and Further Applications of Tuza's Set Pair Method

“…Note that such a family is unextendable not only by any k-subsets of its underlying set, but by any k-sets in the universe at all. This kind of maximal intersecting set systems were studied a lot, the best known upper bound on f (k) is due to Majumder [16], stating…”