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CrossIntersecting Families of Vectors
"... Abstract. Given a sequence of positive integers p = (p1,..., pn), let Sp denote the family of all sequences of positive integers x = (x1,..., xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, are said to be rcrossintersecting if no matter how we select x ∈ A and ..."
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and y ∈ B, there are at least r distinct indices i such that xi = yi. We determine the maximum value of A  · B  over all pairs of rcrossintersecting families and characterize the extremal pairs for r ≥ 1, provided that min pi> r + 1. The case min pi ≤ r + 1 is quite different. For this case
CrossIntersecting Sets of Vectors
"... Given a sequence of positive integers p = (p1,..., pn), let Sp denote the set of all sequences of positive integers x = (x1,..., xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A,B ⊆ Sp, are said to be rcrossintersecting if no matter how we select x ∈ A and y ∈ B, there ar ..."
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, there are at least r distinct indices i such that xi = yi. We show that for any pair of 1crossintersecting families, A,B ⊆ Sp, we have A·B  ≤ Sp2/k2, where k = mini pi. We also determine the minimal value of A  · B  for any pair of rcrossintersecting families and characterize the extremal pairs for r
On families of weakly crossintersecting setpairs
"... Let F be a family of pairs of sets. We call it an (a, b)set system if for every setpair (A,B) in F we have that A  = a, B  = b, A ∩ B = ∅. The following classical result on families of crossintersecting setpairs is due to Bollobás [6]. Let F be an (a, b)set system with the crossintersec ..."
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Let F be a family of pairs of sets. We call it an (a, b)set system if for every setpair (A,B) in F we have that A  = a, B  = b, A ∩ B = ∅. The following classical result on families of crossintersecting setpairs is due to Bollobás [6]. Let F be an (a, b)set system with the crossintersecting
Multiple crossintersecting families of signed sets
 J. Combin. Theory Ser. A
, 2010
"... Abstract A ksigned rset on [n] = {1, ..., n} is an ordered pair (A, f ), where A is an rsubset of [n] and f is a function from A to [k]. Families A 1 , ..., A p are said to be crossintersecting if any set in any family A i intersects any set in any other family A j . Hilton proved a sharp bound ..."
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Cited by 8 (3 self)
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Abstract A ksigned rset on [n] = {1, ..., n} is an ordered pair (A, f ), where A is an rsubset of [n] and f is a function from A to [k]. Families A 1 , ..., A p are said to be crossintersecting if any set in any family A i intersects any set in any other family A j . Hilton proved a sharp bound
Almost crossintersecting and almost crossSperner pairs of families of sets
 Graphs Combin
"... Abstract For a set G and a family of sets F let D ..."
Uniformly cross intersecting families
, 2006
"... Let A and B denote two families of subsets of an nelement set. The pair (A, B) is said to be ℓcrossintersecting iff A∩B  = ℓ for all A ∈ A and B ∈ B. Denote by Pℓ(n) the maximum value of AB  over all such pairs. The best known upper bound on Pℓ(n) is Θ(2n), by Frankl and Rödl. For a lower ..."
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Cited by 1 (0 self)
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Let A and B denote two families of subsets of an nelement set. The pair (A, B) is said to be ℓcrossintersecting iff A∩B  = ℓ for all A ∈ A and B ∈ B. Denote by Pℓ(n) the maximum value of AB  over all such pairs. The best known upper bound on Pℓ(n) is Θ(2n), by Frankl and Rödl. For a lower
Crossing families
 Combinatorica
, 1994
"... Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two sets A and B of points in the plane are mutually avoiding if no line subtended by a pair of points in A intersects the con ..."
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Cited by 17 (2 self)
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Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two sets A and B of points in the plane are mutually avoiding if no line subtended by a pair of points in A intersects
CrossDisjoint Pairs Of Clouds In The Interval Lattice
 ALGORITHMS AND COMBINATORICS B
, 1996
"... Let I n be the lattice of intervals in the Boolean lattice L n . For A; B ae I n the pair of clouds (A; B) is crossdisjoint, if I " J = OE for I 2 A , J 2 B . We prove that for such pairs jAjjBj 3 2n\Gamma2 and that this bound is best possible. Optimal pairs are up to obvious isomorphisms u ..."
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Cited by 3 (1 self)
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unique. The proof is based on a new bound on cross intersecting families in L n with a weight distribution. It implies also an Intersection Theorem for multisets of Erdős and Schönheim [9].
On the RichterThomassen Conjecture about Pairwise Intersecting Closed Curves
, 2014
"... A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane which are in general position and any pair of them intersect, is at least (1 − o(1))n2. We confirm the above conjecture in several important ..."
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in common at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on R, no three of which pass through the same point. If there are nt pairs of tangent curves in S, then the number
Plumbing Graphs for Normal SurfaceCurve Pairs
"... Consider the set of surfacecurve pairs (X; C), where X is a normal surface and C is an algebraic curve. In this paper, we dene a family F of normal surfacecurve pairs, which is closed under coverings, and which contains all smooth surfacecurve pairs (X; C), where X is smooth and C has smooth irr ..."
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Consider the set of surfacecurve pairs (X; C), where X is a normal surface and C is an algebraic curve. In this paper, we dene a family F of normal surfacecurve pairs, which is closed under coverings, and which contains all smooth surfacecurve pairs (X; C), where X is smooth and C has smooth
Results 1  10
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