On the shadow of squashed families of k-sets
نویسنده
چکیده
The shadow of a collection A of k-sets is defined as the collection of the (k − 1)-sets which are contained in at least one k-set of A. Given |A|, the size of the shadow is minimum when A is the family of the first k-sets in squashed order (by definition, a k-set A is smaller than a k-set B in the squashed order if the largest element of the symmetric difference of A and B is in B). We give a tight upper bound and an asymptotic formula for the size of the shadow of squashed families of k-sets. Submitted: January 15, 1995; Accepted: August 25, 1995. AMS Subject Classification. 04A20,03E05,05A20.
منابع مشابه
$Z_k$-Magic Labeling of Some Families of Graphs
For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic} if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$ the group of integers modulo $k...
متن کامل12 Rudolf Ahlswede ,
One of the basic results in extremal set theory was discovered in [1] and rediscovered in [2]: For a given number of k-element subsets of an n-set the shadow, that is, the set of ( k 1)-element subsets contained in at least one of the specified k-element subsets, is minimal, if the k-element subsets are chosen as an initial segment in the squashed order (see [10]; called colex order in [liD, th...
متن کاملNonexistence of a Kruskal–Katona type theorem for double-sided shadow minimization in the Boolean cube layer
A double-sided shadow minimization problem in the Boolean cube layer is investigated in this paper. The problem is to minimize the size of the union of the lower and upper shadows of a k-uniform family of subsets of [n]. It is shown that if 3 ≤ k ≤ n− 3, there is no total order such that all its initial segments have minimal double-sided shadow. Denote by ([n] k ) the family of all subsets of t...
متن کاملStrength of strongest dominating sets in fuzzy graphs
A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...
متن کاملAntichains on Three Levels
An antichain is a collection of sets in which no two sets are comparable under set inclusion. An antichain A is flat if there exists an integer k ≥ 0 such that every set in A has cardinality either k or k + 1. The size of A is |A| and the volume of A is ∑ A∈A |A|. The flat antichain theorem states that for any antichain A on [n] = {1, 2, . . . , n} there exists a flat antichain on [n] with the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 2 شماره
صفحات -
تاریخ انتشار 1995