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arXiv · 1309.4643

Set Systems Containing Many Maximal Chains

J. Robert Johnson·Imre Leader·Paul A. Russell
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Abstract

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $α>0$ there is a family of $α2^n$ sets containing $(α+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. 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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BibTeXRIS

J. Robert Johnson, Imre Leader, Paul A. Russell. 2013-09-18. Set Systems Containing Many Maximal Chains. https://doi.org/10.1017/s0963548314000510

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