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Set Systems Containing Many Maximal Chains

  • J. ROBERT JOHNSON (a1), IMRE LEADER (a2) and PAUL A. RUSSELL (a2)

Abstract

The purpose of this short problem paper is to raise the following extremal question on set systems: Which set systems of a given size maximise the number of (n + 1)-element chains in the power set $\mathcal{P}$ (1,2,. . .,n)? We will show that for each fixed α > 0 there is a family of α2n sets containing (α + o(1))n! such chains, and that this is asymptotically best possible. For smaller set systems we conjecture that a ‘tower of cubes’ construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.

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[1]Alon, N. and Frankl, P. (1985) The maximum number of disjoint pairs in a family of subsets. Graphs Combin. 1 1321.
[2]Bollobás, B. (1986) Combinatorics, Cambridge University Press.
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[4]Dove, A. P., Griggs, J. R., Kang, R. J. and Sereni, J.-S. (2014) Supersaturation in the Boolean lattice. Integers 14A, Paper No. A4
[5]Katona, G. O. H., Katona, G. Y. and Katona, Z. (2012) Most probably intersecting families of subsets. Combin. Probab. Comput. 21 219227.
[6]Kleitman, D. (1968) A conjecture of Erdős–Katona on commensurable pairs of subsets of an n-set. In Theory of Graphs: Proc. Colloquium Held at Tihany, Hungary, September 1966 (Erdős, P. and Katona, G., eds), Academic Press, pp. 215218.
[7]Russell, P. A. (2012) Compressions and probably intersecting families. Combin. Probab. Comput. 21 301313.

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