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

Article Type

Research Article

Publication Title

Combinatorica

Abstract

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.

First Page

87

Last Page

97

DOI

10.1007/s00493-015-3275-8

Publication Date

2-1-2017

Comments

Open Access, Green

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