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Extremal Problems in Partially Ordered Sets

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Ordered Sets

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 83))

Abstract

The central problem among extremal problems in partially ordered sets (posets) is to find the largest set of mutually incomparable elements in a poset. The first results were obtained by Sperner in 1928 [Sr]: the maximum-sized set of mutually incomparable subsets of an n-element set consists of those subsets with [n/2] elements, or those with [n/2] elements. A completely different approach was taken by Dilworth in 1950 [Di]: the size of the largest incomparable collection (antichain) equals the smallest number of totally ordered collections (chains) whose union includes all of the elements of the poset.

Sperner’s theorem is typical of attempts to characterize extremal configurations explicitly for a class of posets. Dilworth’s theorem applies, to all posets but gives a less “precise” result: it gives only the maximum size (although many algorithms which compute that will also find a maximum collection).

Both of these theorems have been proved in many ways and, especially in recent years, both have been generalized in many ways. We also consider several other extremal problems, some arising from the correspondence between the antichains and the order ideals of a poset.

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  1. Department of Mathematics, Princeton University, Princeton, New Jersey, 08544, USA

    Douglas B. West

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  1. Douglas B. West
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  1. Department of Mathematics and Statistics, The University of Calgary, Canada

    Ivan Rival

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© 1982 D. Reidel Publishing Company

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West, D.B. (1982). Extremal Problems in Partially Ordered Sets. In: Rival, I. (eds) Ordered Sets. NATO Advanced Study Institutes Series, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7798-3_16

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  • DOI: https://doi.org/10.1007/978-94-009-7798-3_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-7800-3

  • Online ISBN: 978-94-009-7798-3

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