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 \(N\left( k \right) \geqslant 2k - 2 + \frac{1}{2}\left( {\begin{array}{*{20}{c}} {2k - 2} \\ {k - 1} \end{array}} \right)\). 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.

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Correspondence to Kaushik Majumder.

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Majumder, K. On the maximum number of points in a maximal intersecting family of finite sets. Combinatorica 37, 87–97 (2017). https://doi.org/10.1007/s00493-015-3275-8

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Mathematics Subject Classification (2000)

  • 05D05
  • 05D15
  • 05C65
  • 05A16