On Saturated k-Sperner Systems
نویسندگان
چکیده
Given a set X, a collection F ⊆ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P(X) is 2k−1. In this talk we disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n0(k) there exists a saturated k-Sperner system F ⊆ P(X) with cardinality at most 2(1−ε)k.
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عنوان ژورنال:
- Electr. J. Comb.
دوره 21 شماره
صفحات -
تاریخ انتشار 2014