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On Lih's Conjecture concerning Spernerity

✍ Scribed by D.G.C. Horrocks

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
237 KB
Volume
20
Category
Article
ISSN
0195-6698

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✦ Synopsis

Let F be a nonempty collection of subsets of [n] = {1, 2, . . . , n}, each having cardinality t. Denote by P F the poset consisting of all subsets of [n] which contain at least one member of F , ordered by set-theoretic inclusion. In 1980, K. W. Lih conjectured that P F has the Sperner property for all 1 ≀ t ≀ n and every choice of F . This conjecture is known to be true for t = 1 but false, in general, for t β‰₯ 4. In this paper, we prove Lih's conjecture in the case t = 2.

We make extensive use of fundamental theorems concerning the preservation of Sperner-type properties under direct products of posets.

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