On the structure of maximum 2-part Sperner families
نویسنده
چکیده
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 structure and number of all maximum 2-part Sperner families, using only some elementary properties of symmetric chain decompositions of the poset of all subsets of a finite set.
منابع مشابه
All Maximum Size Two-Part Sperner Systems: In Short
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...
متن کاملConvex hulls of more-part Sperner families
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 کاملThe Linear Chromatic Number of Sperner Families
Let S be a set with m elements and S a complete Sperner family on S, i.e. a Sperner family such that every x ∈ S is contained in some member of S. The linear chromatic number of S, defined by Civan, is the smallest integer n with the property that there exists a function f : S → {1, . . . , n} such that if f(x) = f(y), then every set in S which contains x also contains y, or every set in S whic...
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 ...
متن کاملThe linear chromatic number of a Sperner family
Let S be a finite set and S a complete Sperner family on S, i.e. a Sperner family such that every x ∈ S is contained in some member of S. The linear chromatic number of S, defined by Cıvan, is the smallest integer n with the property that there exists a function f : S → {1, . . . , n} such that if f(x) = f(y), then every set in S which contains x also contains y or every set in S which contains...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 162 شماره
صفحات -
تاریخ انتشار 1996