All maximum 2-part Sperner families
نویسندگان
چکیده
منابع مشابه
All maximum 2-part Sperner families
Let X= X, v ,I’?, X, n X2 = 0 be a partition of an n-element set. Suppose that the family % of some subsets of X satisfy the following condition: if there is an inclusion F, G Fz (F,, Fz E 5”) in %, the difference F, F, cannot be a subset of X, or ,I’>. Kleitman (Math. Z. 90 (1965), 251-259) and Katona (Sfudia Sci. Math. Hungar. 1 (1966) 59-63) proved 20 years ago that 1% 1 is at most n choose ...
متن کاملOn the structure of maximum 2-part Sperner families
Color the elements of a finite set S with two colors. A collection of subsets of S is called a 2-part Sperner family if whenever for two distinct sets A and B in this collection we have A ⊂ B then B − A has elements of S of both colors. All 2-part Sperner families of maximum size were characterized in Erdös and Katona [5]. In this paper we provide a different, and quite elementary proof of the ...
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Katona [6] and Kleitman [8] independently observed that the statement of the Sperner theorem remains unchanged if the conditions are relaxed in the following way. Let X = X1 ∪X2 be a partition of the underlying set X, |Xi| = ni, n1 + n2 = n (with n1 n2). We say that F is a two-part Sperner family if E, F ∈ F, E F ⇒ ∀i : (F \ E) ⊂ Xi. It was proved that the size of a two-part Sperner family cann...
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The convex hulls of more-part Sperner families is defined and studied. Corollaries of the results are some well-known theorems on 2 or 3-part Sperner families. Some methods are presented giving new theorems.
متن کاملTwo-Part and k-Sperner Families: New Proofs Using Permutations
This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdős and G. O. H. Katona, J. Combin. Theory. Ser. A, 43 (1986), pp. 58–69; S. Shahriari, Discrete Math., 162 (1996), pp. 229–238] determining all extremal two-part Sperner families and for the uniqueness of k-Sperner families of maximum size [P. Erdős, Bull. Amer. Math. Soc., 51 ...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1986
ISSN: 0097-3165
DOI: 10.1016/0097-3165(86)90023-3