On Lih's Conjecture concerning Spernerity
نویسنده
چکیده
LetF be a nonempty collection of subsets of [n] = {1, 2, . . . , n}, each having cardinality t . Denote by PF 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 PF 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 ≥ 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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عنوان ژورنال:
- Eur. J. Comb.
دوره 20 شماره
صفحات -
تاریخ انتشار 1999