Combinatorica

, Volume 37, Issue 1, pp 87–97 | Cite as

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

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

Mathematics Subject Classification (2000)

05D05 05D15 05C65 05A16 

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References

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    B. BollobÁs: On generalized graphs, Acta Mathematica Academiae Scientiarum Hun-garicae 16 (1965), 447–452.CrossRefMATHGoogle Scholar
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    P. Erdős and L. LovÁsz: Problems and results on 3-chromatic hypergraphs and some related questions, Infinite and finite sets (Proceedings of a Colloquium held at Keszthely from June 25 to July 1;1973. Dedicated to Paul Erdős on his 60th birth-day), Volume II, North-Holland, Amsterdam, 1975, 609–627. Colloquia MathematicaSocietatis János Bolyai, Volume 10.Google Scholar
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    D. Hanson and B. Toft: On the maximum number of vertices in n-uniform cliques, Ars Combinatoria 16 (1983), 205–216.MathSciNetMATHGoogle Scholar
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    Zs. Tuza: Critical hypergraphs and intersecting set-pair systems, Journal of Combi-natorial Theory Series B 39 (1985), 134–145.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia

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