Abstract
In this chapter we consider a surprising application of certain adjacency matrices to some problems in extremal set theory. An important role will also be played by finite groups in Chapter 5, which is a continuation of the present chapter. In general, extremal set theory is concerned with finding (or estimating) the most or least number of sets satisfying given set-theoretic or combinatorial conditions. For example, a typical easy problem in extremal set theory is the following: what is the most number of subsets of an n-element set with the property that any two of them intersect? (Can you solve this problem?) The problems to be considered here are most conveniently formulated in terms of partially ordered sets or posets for short. Thus we begin with discussing some basic notions concerning posets.
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References
I. Anderson, Combinatorics of Finite Sets (Oxford University Press, Oxford, 1987); Corrected republication by Dover, New York, 2002
N. Caspard, B. Leclerc, B. Monjardet, Finite Ordered Sets. Encyclopedia of Mathematics and Its Applications, vol. 144 (Cambridge University Press, Cambridge, 2012)
K. Engel, Sperner Theory. Encyclopedia of Mathematics and Its Applications, vol. 65 (Cambridge University Press, Cambridge, 1997)
P. Fishburn, Interval Orders and Interval Graphs: A Study of Partially Ordered Sets (Wiley, New York, 1985)
D. Lubell, A short proof of Sperner’s lemma. J. Comb. Theor. 1, 299 (1966)
M. Pouzet, Application d’une propriété combinatoire des parties d’un ensemble aux groupes et aux relations. Math. Zeit. 150, 117–134 (1976)
M. Pouzet, I.G. Rosenberg, Sperner properties for groups and relations. Eur. J. Combin. 7, 349–370 (1986)
E. Sperner, Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27(1), 544–548 (1928)
R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebr. Discrete Meth. 1, 168–184 (1980)
R. Stanley, Quotients of Peck posets. Order 1, 29–34 (1984)
R. Stanley, Enumerative Combinatorics, vol. 1, 2nd edn. (Cambridge University Press, Cambridge, 2012)
W.T. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins Studies in the Mathematical Sciences, vol. 6 (Johns Hopkins University Press, Baltimore, 1992)
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Stanley, R.P. (2018). The Sperner Property. In: Algebraic Combinatorics. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77173-1_4
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DOI: https://doi.org/10.1007/978-3-319-77173-1_4
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