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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\left( k \right) \geqslant 2k - 2 + \frac{1}{2}\left( {\begin{array}{*{20}{c}} {2k - 2} \\ {k - 1} \end{array}} \right)$$\end{document}. 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.