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A Stability Result for Families with Fixed Diameter

Published online by Cambridge University Press:  06 March 2017

PETER FRANKL*
Affiliation:
Alfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13--15, H-1053, Budapest, Hungary (e-mail: [email protected])

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

Type
Paper
Copyright
Copyright © Cambridge University Press 2017 

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References

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