Abstract
This paper is a summary (without proofs) of the main results in a series of papers by the author and D.J. Kleitman [14] and the author [11, 12, 13] concerning subsets of a finite partially ordered set called Sperner k-families. If P is a finite partially ordered set, a subset A ⊆ P is a k-family if A contains no chains of length k+1 (or, equivalently, if A can be expressed as the union of k 1-families in P). Maximum-sized k-families are called Sperner k-families of P.
Supported in part by ONR N00014-67-A-0204-0063.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Berge, C., Färbung von Graphen deren samtliche bzw. ungerade Kreise starr sind (Zusammenfassung), Wiss. Z. Martin Luther Univ. Halle Wittenberg, Math. Nat. Reihe, 1961, pp. 114.
Birkhoff, G., Lattice theory, AMS Coll. Publ. 25, Amer. Math. Soc., Providence, 1967.
Brualdi, R., Induced matroids, Proc. Amer. Math. Soc., 29 (1971) 213–221.
Dilworth, R.P., A decomposition theorem for partially ordered sets, Ann. of Math., 51 (1950) 161–166.
Dilworth, R.P., Some combinatorial problems on partially ordered sets, in: Combinatorial analysis, R. Bellman & M. Hall Jr. (eds.), Proc. Symp. Appl. Math., Amer. Math. Soc., Providence, 1960, pp. 85–90.
Edmonds, J., Submodular functions, matroids and certain polyhedra, in: Combinatorial structures and their applications, Proc. Calgary Internat. Conference 1969, Gordon and Breach, New York, 1970, pp. 69.
Erdös, P., On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945) 898–902.
Freese, R., An application of Dilworth’s lattice of maximal antichains, Discrete Math., 7 (1974) 107–109.
Fulkerson, D.R., A note on Dilworth’s theorem for partially ordered sets, Proc. Amer. Math. Soc., 7 (1956) 701–702.
Fulkerson, D.R., Anti-blocking polyhedra, J. Combinatorial Theory B, 12 (1972) 50–71.
Greene, C., An extension of Schensted’s theorem, to appear.
Greene, C., Partitions and conjugate partitions of a partially ordered set, to appear.
Greene, C., Construction of Sperner k-families and k-saturated partitions, to appear.
Greene, C. & D.J. Kleitman, On the structure of Sperner k-families, to appear.
Kleitman, D.J., M. Edelberg & D. Lubell, Maximal sized antichains in partial orders, Discrete Math., 1 (1971) 47–53.
Lovász, L., Normal hypergraphs and the perfect graph conjecture, Discrete Math., 2 (1972) 253–267.
Schensted, C., Longest increasing and decreasing subsequences, Canad. J. Math., 13 (1961) 179–191.
Sperner, E., Ein Satz über Untermengen einer endlichen Menge, Math. Z., 27 (1928) 544–548.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1975 Mathematical Centre, Amsterdam
About this paper
Cite this paper
Greene, C. (1975). Sperner Families and Partitions of A Partially Ordered Set. In: Hall, M., van Lint, J.H. (eds) Combinatorics. NATO Advanced Study Institutes Series, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1826-5_15
Download citation
DOI: https://doi.org/10.1007/978-94-010-1826-5_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-1828-9
Online ISBN: 978-94-010-1826-5
eBook Packages: Springer Book Archive