The colored Tverberg's problem and complexes of injective functions

Živaljević, Rade T.; Vrecica, Sinša T.

doi:10.1016/0097-3165(92)90028-s

Cited by 124 publications

(73 citation statements)

References 5 publications

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“…Bárány, Füredi and Lovász [4] suggested a colored version of Tverberg's theorem, which was then established byŽivaljević and Vrećica [22] in arbitrary dimensions. The planar case allowed a quantitative improvement, provided by Bárány and Larman [3].…”

Section: Introductionmentioning

confidence: 99%

Algorithms for center and Tverberg points

Agarwal

Sharir

Welzl

2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

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We present a near-quadratic algorithm for computing the center region of a set of n points in three dimensions. This is nearly tight in the worst case since the center region can have Ω(n 2 ) complexity. We then consider the problem of recognizing whether a given point q is a colored Tverberg point of a set of n colored points in the plane, and present the first polynomial-time algorithm for this problem.

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Section: Introductionmentioning

confidence: 99%

Algorithms for center and Tverberg points

Agarwal

Sharir

Welzl

2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

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“…The chessboard complex also appeared in a 1992 paper ofZivaljević and Vrécica [54] in connection with some problems in computational geometry. One of these problems can be described as follows.…”

Section: History and Connectionsmentioning

confidence: 96%

“…Graph complexes have provided an important link between combinatorics and algebra, topology and geometry; see eg., [29,11,47,54,2,48,37]. Here we consider the simplicial complexes associated with the collection of subgraphs of a (di-, multi-, hyper-)graph G whose node degrees are bounded from above.…”

Section: Introductionmentioning

confidence: 99%

Topology of matching, chessboard, and general bounded degree graph complexes

Wachs

2003

Algebra Universalis

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Abstract. We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.

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“…• For hyperplanes in a fixed dimension d > 3, Bárány et al's proof in combination with the multicolored Tverberg theorem [29,6] yielded an upper bound of O(n d−α d ) for a very small α d = 1/(2d) d . As Agarwal et al [1] observed, the bound can be made sensitive to k, namely, O(n d/2 k d/2 −α d ).…”

Section: Introductionmentioning

confidence: 99%

On Levels in Arrangements of Surfaces in Three Dimensions

Chan

2012

Discrete Comput Geom

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A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n "pseudo-planes" or "pseudo-spherical patches" (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n 2.997 ) vertices at any given level.

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The colored Tverberg's problem and complexes of injective functions

Cited by 124 publications

References 5 publications

Algorithms for center and Tverberg points

Algorithms for center and Tverberg points

Topology of matching, chessboard, and general bounded degree graph complexes

On Levels in Arrangements of Surfaces in Three Dimensions

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