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A Stability Result for Families with Fixed Diameter

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  06 March 2017

PETER FRANKL
PETER FRANKL*
Affiliation:
Alfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13--15, H-1053, Budapest, Hungary (e-mail: frankl.peter@renyi.mta.hu)
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Abstract

Let $\mathcal F$ ⊂ 2[n] be a family of subsets. The diameter of $\mathcal F$ is the maximum of the size of symmetric differences among pairs of members of $\mathcal F$. In 1966 Kleitman determined the maximum of |$\mathcal F$| for fixed diameter. However, this important classical result lacked a characterization of the families meeting the bound. This is remedied in the present paper, where a best possible stability result is established as well.

In Section 4 we introduce a ‘parity trick’ that provides an easy way of deducing the odd case from the even case in both Kleitman's original theorem and its stability version.

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 506 - 516
Copyright
Copyright © Cambridge University Press 2017 

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References

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