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Three famous theorems on finite sets

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Proofs from THE BOOK

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Abstract

In this chapter we are concerned with a basic theme of combinatorics: properties and sizes of special families F of subsets of a finite set N = {1, 2,..., n}. We start with two results which are classics in the field: the theorems of Sperner and of Erdős-Ko-Rado. These two results have in common that they were reproved many times and that each of them initiated a new field of combinatorial set theory. For both theorems, induction seems to be the natural method, but the arguments we are going to discuss are quite different and truly inspired.

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References

  1. T. E. Easterfield: A combinatorial algorithm, J. London Math. Soc. 21 (1946), 219–226.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Erdős, C. KO & R. Rado: Intersection theorems for systems of finite sets, Quart. J. Math. (Oxford), Ser. (2) 12 (1961), 313–320.

    Article  MathSciNet  Google Scholar 

  3. P. Hall: On representatives of subsets, Quart. J. Math. (Oxford) 10 (1935), 26–30.

    Google Scholar 

  4. P. R. Halmos & H. E. Vaughan: The marriage problem, Amer. J. Math. 72 (1950), 214–215.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Katona: A simple proof of the Erdos-Ko-Rado theorem, J. Combinatorial Theory, Ser. B13 (1972), 183–184.

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  6. L. Lovász & M. D. Plummer: Matching Theory, Akadémiai Kiadö, Budapest 1986.

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  7. D. Lubell: A short proof of Sperner’s theorem, J. Combinatorial Theory 1 (1966), 299.

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  8. E. Sperner: Ein Satz über Untermengen einer endlichen Menge, Math. Zeitschrift 27 (1928), 544–548.

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Authors and Affiliations

  1. Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Martin Aigner

  2. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

Authors
  1. Martin Aigner
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  2. Günter M. Ziegler
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© 2001 Springer-Verlag Berlin Heidelberg

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Aigner, M., Ziegler, G.M. (2001). Three famous theorems on finite sets. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04315-8_22

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  • DOI: https://doi.org/10.1007/978-3-662-04315-8_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-04317-2

  • Online ISBN: 978-3-662-04315-8

  • eBook Packages: Springer Book Archive

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