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11
Crossing patterns of semialgebraic sets
 J. Combin. Theory Ser. A
, 2005
"... We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 int ..."
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Cited by 17 (6 self)
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We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 intersects all elements of F2 or no element of F1 intersects any element of F2. This implies the existence of another constant δ such that F has a subset F ′ ⊆ F with n δ elements, so that either every pair of elements of F ′ intersect each other or the elements of F ′ are pairwise disjoint. The same results hold when the intersection relation is replaced by any other semialgebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory. 1
Crossing Patterns of Segments
, 2001
"... It is shown that for every c > 0 there exists c > 0 satisfying the following condition. Let S be a system of n straightline segments in the plane, which determine at least cn crossings. Then there are two disjoint at least c nelement subsystems, S 1 ; S 2 S, such that every element o ..."
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Cited by 12 (7 self)
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It is shown that for every c > 0 there exists c > 0 satisfying the following condition. Let S be a system of n straightline segments in the plane, which determine at least cn crossings. Then there are two disjoint at least c nelement subsystems, S 1 ; S 2 S, such that every element of S 1 crosses all elements of S 2 .
Ramseytype theorem for bipartite graphs, Geombinatorics 10
, 2000
"... Let H be a fixed graph with k vertices. It is proved that every graph G with n vertices, which does not contain an induced subgraph isomorphic to H, has two disjoint sets of vertices, V1, V2 ∈ V (G), such that V1, V2  ≥ ⌊(n/k) 1/(k−1) ⌋ and either all edges between V1 and V2 belong to G or non ..."
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Cited by 11 (6 self)
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Let H be a fixed graph with k vertices. It is proved that every graph G with n vertices, which does not contain an induced subgraph isomorphic to H, has two disjoint sets of vertices, V1, V2 ∈ V (G), such that V1, V2  ≥ ⌊(n/k) 1/(k−1) ⌋ and either all edges between V1 and V2 belong to G or none of them does. Some related geometric questions are also discussed. 1
Betti Number Bounds, Applications and Algorithms
 Current Trends in Combinatorial and Computational Geometry: Papers from the Special Program at MSRI, MSRI Publications Volume 52
, 2005
"... Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such ..."
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Cited by 8 (5 self)
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Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved useful in several applications in theoretical computer science and discrete geometry. The main goal of this survey paper is to provide an up to date account of the known bounds on the Betti numbers of semialgebraic sets in terms of various parameters, sketch briefly some of the applications, and also survey what is known about the complexity of algorithms for computing them. 1.
Edges and Switches, Tunnels and Bridges
"... Abstract. Edge casing is a wellknown method to improve the readability of drawings of nonplanar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than othe ..."
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Cited by 3 (1 self)
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Abstract. Edge casing is a wellknown method to improve the readability of drawings of nonplanar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a “good ” cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NPhardness results. 1
A RamseyType Theorem for Bipartite Graphs
"... Let H be a fixed graph with k vertices. It is proved that every graph G with n vertices, which does not contain an induced subgraph isomorphic to H , has two disjoint sets of vertices, V 1 , V 2 V (G), such that , V  # #(n/k) and either all edges between V 1 and V 2 belong to G or ..."
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Cited by 1 (0 self)
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Let H be a fixed graph with k vertices. It is proved that every graph G with n vertices, which does not contain an induced subgraph isomorphic to H , has two disjoint sets of vertices, V 1 , V 2 V (G), such that , V  # #(n/k) and either all edges between V 1 and V 2 belong to G or none of them does. Some related geometric questions are also discussed.
Cutting triangular cycles of lines in space
 Proc. 35th Annu. ACM Sympos. Theory Comput
, 2003
"... We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1 ..."
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Cited by 1 (1 self)
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We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1
Spindle configurations of skew lines
, 2002
"... Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists ..."
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Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists. 1.