On Lih's conjecture concerning spernerity

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Title On Lih's conjecture concerning spernerity
Author(s) D. G. C. Horrocks
Journal European Journal of Combinatorics
Date 1999
Volume 20
Issue 2
Start page 131
End page 148
Abstract Let F be a nonempty collection of subsets of [n] = {1, 2,..., n}, each having cardinality t. Denote by P-F the poser 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 less than or equal to t less than or equal to n and every choice of F. This conjecture is known to be true for t = 1 bur false, in general, for t greater than or equal to 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 posers. (C) 1999 Academic Press.

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