Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T05:47:32.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Sperner's Theorem and a Problem of Erdős, Katona and Kleitman

Published online by Cambridge University Press:  01 December 2014

SHAGNIK DAS ,
WENYING GAN  and
BENNY SUDAKOV
SHAGNIK DAS
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: shagnik@ucla.edu, wgan@math.ucla.edu)
WENYING GAN
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: shagnik@ucla.edu, wgan@math.ucla.edu)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zürich, Switzerland (e-mail: benjamin.sudakov@math.ethz.ch)
Get access Rights & Permissions [Opens in a new window]

Abstract

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain, F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain, F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.

In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

Keywords

Primary 05D05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 4: Oberwolfach Special Issue Part 1 , July 2015 , pp. 585 - 608
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chung, F. R. K., Füredi, Z., Graham, R. L. and Seymour, P. (1988) On induced subgraphs of the cube. J. Combin. Theory Ser. A 49 180–187.CrossRefGoogle Scholar
[2]Das, S., Gan, W. and Sudakov, B. (2014) The minimum number of disjoint pairs in set systems and related problems. Combinatorica, to appear.Google Scholar
[3]Dove, A. P., Griggs, J. R., Kang, R. J. and Sereni, J. S. (2014) Supersaturation in the Boolean lattice. Integers 14A 1–7.Google Scholar
[4]Engel, K. (1997) Sperner Theory, Cambridge University Press.CrossRefGoogle Scholar
[5]Erdős, P. (1945) On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 898–902.CrossRefGoogle Scholar
[6]Erdős, P. (1962) On a theorem of Rademacher–Turán. Illinois J. Math. 6 122–127.CrossRefGoogle Scholar
[7]Erdős, P. (1962) On the number of complete subgraphs contained in certain graphs. Magy. Tud. Acad. Mat. Kut. Int. Közl. 7 459–474.Google Scholar
[8]Erdős, P. and Kleitman, D. (1974) Extremal problems among subsets of a set. Discrete Math. 8 281–294.CrossRefGoogle Scholar
[9]Katona, G. O. H. and Tarján, T. G. (1983) Extremal problems with excluded subgraphs in the n-cube. In Graph Theory (Borowiecki, M.et al., eds), Vol. 1018 of Lecture Notes in Mathematics, Springer, pp. 84–93.Google Scholar
[10]Kleitman, D. (1966) A conjecture of Erdős–Katona on commensurable pairs among subsets of an n-set. In Theory of Graphs: Proc. Colloq. Tihany, pp. 215–218.Google Scholar
[11]Mantel, W. (1907) Problem 28. Wiskundige Opgaven 10 60–61.Google Scholar
[12]Qian, J., Engel, K. and Xu, W. (2009) A generalization of Sperner's theorem and an application to graph orientations. Discrete Appl. Math. 157 2170–2176.CrossRefGoogle Scholar
[13]Sperner, E. (1928) Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27 544–548.CrossRefGoogle Scholar