Results 1  10
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312
Maximal antichains of minimum size
, 2013
"... Let n � 4 be a natural number, and let K be a set K ⊆ [n]: = {1, 2,..., n}. We study the problem to find the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K, and A ⊆ B for all {A, B} ⊆ A, i.e. A is an antichain. We present a general ..."
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Cited by 1 (1 self)
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Let n � 4 be a natural number, and let K be a set K ⊆ [n]: = {1, 2,..., n}. We study the problem to find the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K, and A ⊆ B for all {A, B} ⊆ A, i.e. A is an antichain. We present a general
On crossintersecting families of sets
 Graphs Combin
"... Abstract. A family A of ‘element subsets and a family B of kelement subsets of an nelement set are crossintersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product jAjjBj, a ..."
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Cited by 9 (0 self)
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Abstract. A family A of ‘element subsets and a family B of kelement subsets of an nelement set are crossintersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product j
Maximal flat antichains of minimum weight
"... We study maximal families A of subsets of [n] = {1,2,...,n} such that A contains only pairs and triples and A ̸ ⊆ B for all {A,B} ⊆ A, i.e. A is an antichain. For any n, all such families A of minimum size are determined. This is equivalent to finding all graphs G = (V,E) with V  = n and with t ..."
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Cited by 4 (3 self)
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We study maximal families A of subsets of [n] = {1,2,...,n} such that A contains only pairs and triples and A ̸ ⊆ B for all {A,B} ⊆ A, i.e. A is an antichain. For any n, all such families A of minimum size are determined. This is equivalent to finding all graphs G = (V,E) with V  = n
Intersecting Families of Permutations
 EUROPEAN J. COMBIN
, 2003
"... Let S n be the symmetric group on the set X = ....., n}. A subset S of S n is intersecting if for any two permutations g and h in S, g(x) = h(x) for X (that is g and h agree on x). M. Deza and P. Frankl [4] proved that if S S n is intersecting then (n1)!. This bound is met by taking S to ..."
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Cited by 52 (3 self)
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Let S n be the symmetric group on the set X = ....., n}. A subset S of S n is intersecting if for any two permutations g and h in S, g(x) = h(x) for X (that is g and h agree on x). M. Deza and P. Frankl [4] proved that if S S n is intersecting then (n1)!. This bound is met by taking
Combinatorial problems on subsets and their intersections
 Adv. Math., Suppl. Stud
, 1978
"... Let I S I = n, m(n; k l,k 2,k) respectively m'(n,k 1,k, k) denote the cardinali.ty of the largest family of subsets A i C S satisfying IA i I = k (respectively 1A í 1 S k) and 1A i n Ai 1 = k l or Z 2~ 912 * In this paper we prove a) m(n,0,k 2,k) (n), mI(n,0,k 2,k) 5 (2) + ntl; equality, iff k ..."
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Cited by 1 (0 self)
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Let I S I = n, m(n; k l,k 2,k) respectively m'(n,k 1,k, k) denote the cardinali.ty of the largest family of subsets A i C S satisfying IA i I = k (respectively 1A í 1 S k) and 1A i n Ai 1 = k l or Z 2~ 912 * In this paper we prove a) m(n,0,k 2,k) (n), mI(n,0,k 2,k) 5 (2) + ntl; equality, iff k
Maximizing Antichains in the Cube with Fixed Size of a Shadow
, 1992
"... In the ndimensional cube, we determine the maximum size of antichains having a lower shadow of exactly m elements in the kth level. 1 Problem and Result We consider the ndimensional unit cube as the powerset 2 N of N := f1; 2; : : : ; ng, n 1, ordered by inclusion. Then the kth level, 0 k ..."
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Cited by 3 (0 self)
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, an antichain is a family of pairwise incomparable sets in the cube. Throughout the paper, let A ` 2 N be an antichain with jffi k (A)j = m. The main purpose of this paper is to determine the maximal size of A. Let f(n; k; m) := maxfjAj : A 2 2 N ; A antichain ; jffi k (A)j = mg : We repeat the famous
Intersecting families in a subset of boolean lattices
"... Let n,r and ℓ be distinct positive integers with r < ℓ ≤ n/2, and let X1 and X2 be two disjoint sets with the same size n. Define X F = A ∈ : A ∩ X1  = r or ℓ, r + ℓ where X = X1 ∪ ( X2. In) ( this) paper, ( we) ( prove) that if S is an intersecting family n − 1 n n − 1 n in F, then S  ≤ ..."
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Cited by 1 (0 self)
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Let n,r and ℓ be distinct positive integers with r < ℓ ≤ n/2, and let X1 and X2 be two disjoint sets with the same size n. Define X F = A ∈ : A ∩ X1  = r or ℓ, r + ℓ where X = X1 ∪ ( X2. In) ( this) paper, ( we) ( prove) that if S is an intersecting family n − 1 n n − 1 n in F, then S
Intersecting Families with Minimum Volume
, 2001
"... We determine the minimum volume (sum of cardinalities) of an intersecting family of subsets of an nset, given the size of the family, by solving a simple linear program. From this we obtain a lower bound on the average size of the sets in an intersecting family. This answers a question of G. O. H. ..."
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We determine the minimum volume (sum of cardinalities) of an intersecting family of subsets of an nset, given the size of the family, by solving a simple linear program. From this we obtain a lower bound on the average size of the sets in an intersecting family. This answers a question of G. O. H
Intersecting families of permutations
 Journal of the American Mathematical Society
"... A set of permutations I ⊂ Sn is said to be kintersecting if any two permutations in I agree on at least k points. We show that for any k ∈ N, if n is sufficiently large depending on k, then the largest kintersecting subsets of Sn are cosets of stabilizers of k points, proving a conjecture of Deza ..."
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Cited by 27 (6 self)
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A set of permutations I ⊂ Sn is said to be kintersecting if any two permutations in I agree on at least k points. We show that for any k ∈ N, if n is sufficiently large depending on k, then the largest kintersecting subsets of Sn are cosets of stabilizers of k points, proving a conjecture of Deza
Some Intersection Theorems for Ordered Sets and Graphs
 IOURNAL OF COMBINATORIAL THEORY, SERIES A 43, 2337
, 1986
"... A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of a ..."
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Cited by 86 (1 self)
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A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F
Results 1  10
of
312