On the maximum number of points in a maximal intersecting family of finite sets
Paul Erdős and LászlÓ Lovász proved in a landmark article that, for any positive integerk, 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+12(2k−2k−1). Much later, Zsolt Tuza proved that the bound is best possibleup 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 boundby showing that asymptotically N(k) is at most 3 times the Erdős-Lovősz lower bound.A related conjecture of Zsolt Tuza, if proved, would imply that the explicit upper boundobtained in this paper is only double the Erdős-Lovász lower bound.
Majumder, Kaushik, "On the maximum number of points in a maximal intersecting family of finite sets" (2017). Journal Articles. 2712.