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A Stability Result for Families with Fixed Diameter

Part of: Graph theory Extremal combinatorics

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

[1] Frankl, P. A stability result for the Katona Theorem. J. Combin. Theory Ser. B, submitted.Google Scholar
[2] Frankl, P. and Tokushige, N. (1992) Some best possible inequalities concerning cross-intersecting families. J. Combin. Theory Ser. A 61 87–97.CrossRefGoogle Scholar
[3] Hilton, A. J. W. and Milner, E. C. (1967) Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford 18 369–384.CrossRefGoogle Scholar
[4] Katona, G. O. H. (1964) Intersection theorems for systems of finite sets. Acta Math. Hungar. 15 329–337.Google Scholar
[5] Kleitman, D. J. (1966) On a combinatorial conjecture of Erdős. J. Combin. Theory 1 209–214.CrossRefGoogle Scholar
[6] Wang, J. and Zhang, H. (2013) Nontrivial independent sets of bipartite graphs and cross-intersecting families, J. Combin. Theory Ser. A 120 129–141.Google Scholar