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A Slight Improvement to the Colored Bárány's Theorem
[article]
2014
arXiv
pre-print
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In this paper, we prove that there exists a point of R^d that belongs to the convex hull of d+1 points v_0, ..., v_d with probability at least 2d/(d+1)! ...
(d+1), where each point v_i is sampled independently according to probability measure m_i. ...
Acknowledgment The author would like to thank Boris Bukh for guidance and fruitful discussions on the Bárány's theorem. This article would not have been possible without his support. ...
arXiv:1405.2503v2
fatcat:4deh2fsutjgjvfwafvaxh4dsqy
A Slight Improvement to the Colored Bárány's Theorem
unpublished
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In this paper, we prove that there exists a point of R d that belongs to the convex hull of d + 1 points v 0 ,. .. , v d with probability at least 2d (d+1)! ...
(d+1) , where each point v i is sampled independently according to probability measure m i . ...
Acknowledgment The author would like to thank Boris Bukh for guidance and fruitful discussions on the Bárány's theorem. This article would not have been possible without his support. ...
fatcat:2xj2tm4zgngonnzocahferw2ha
Colorful linear programming, Nash equilibrium, and pivots
[article]
2016
arXiv
pre-print
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In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. ...
We also propose a variant of the Bárány algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. ...
The authors thank the reviewers for their helpful comments. ...
arXiv:1409.3436v3
fatcat:ghi6me4jxzhbbemup263pwas7i
On random subgraphs of Kneser and Schrijver graphs
[article]
2015
arXiv
pre-print
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A famous result due to Lovász states that the chromatic number of a Kneser graph KG_n,k is equal to n-2k+2. ...
A Kneser graph KG_n,k is a graph whose vertices are in one-to-one correspondence with k-element subsets of [n], with two vertices connected if and only if the corresponding sets do not intersect. ...
In the paper [15] , Schrijver noted that a slight modification of Bárány's proof allow to prove a much stronger statement: χ(SG n,k ) = n − 2k + 2. ...
arXiv:1502.00699v2
fatcat:jfjwzkjjebdejoye75qvqedlqe
On a topological version of Pach's overlap theorem
[article]
2017
arXiv
pre-print
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This improves upon the results of Bárány, Meshulam, Nevo and Tancer. ...
We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C( n)^1/(d-1). ...
[20] and Bárány's overlap theorem [4, Theorem 5 .1] respectively. ...
arXiv:1708.04350v1
fatcat:vuqg232sjbebpbljs2rzbb4dbe
An optimal generalization of the Colorful Carathéodory theorem
2016
Discrete Mathematics
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The Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with ...
One of our results is the following extension of the Colorful Carathéodory theorem: given d/2 + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant ...
We would also like to thank János Pach for his support and encouragement. ...
doi:10.1016/j.disc.2015.11.019
fatcat:ux52sv7gxvh5zp7hair7dhitgi
A theorem of bárány revisited and extended
2012
Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12
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The colorful Carathéodory theorem [Bár82] states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most ...
One of our results is the following extension of the colorful Carathéodory theorem: given ⌊d/2⌋ + 1 sets of points in R d , and a convex object C, then either one set can be separated from C by a constant ...
The first author would also like to thank Janos Pach for his support and encouragement. ...
doi:10.1145/2261250.2261300
dblp:conf/compgeom/MustafaR12
fatcat:6ttq7n37yrb33inhnc5bhfditq
Fair division and generalizations of Sperner- and KKM-type results
[article]
2017
arXiv
pre-print
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Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. ...
For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. ...
,d C i π(i) = ∅. theorem is "colorful" in the same sense that Bárány's extension of Carathéodory's theorem [8] is a colorful Carathéodory's theorem. ...
arXiv:1701.04955v1
fatcat:6m5zxxh4f5dknc4aef7yzkihni
Random Kneser graphs and hypergraphs
[article]
2018
arXiv
pre-print
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This allows us to improve all known results on the topic. ...
A famous result due to Lovász states that the chromatic number of KG_n,k is equal to n-2k+2. In this paper we discuss the chromatic number of random Kneser graphs and hypergraphs. ...
Schrijver noticed that a slight modification of Bárány's proof yields a stronger statement: χ(SG n,k ) = n − 2k + 2. ...
arXiv:1612.03868v3
fatcat:ghdrlmw4hfgwjklfwo4bben4hq
Random Kneser Graphs and Hypergraphs
2018
Electronic Journal of Combinatorics
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Moreover, they have improved the bounds of Kupavskii in the graph case for many values of parameters.In the present paper we present a purely combinatorial approach to the problem based on blow-ups of ...
This allows us to improve all known results on the topic. ...
Schrijver noticed that a slight modification of Bárány's proof yields a stronger statement: χ(SG n,k ) = n − 2k + 2. ...
doi:10.37236/8005
fatcat:so6rixj5l5ba7lbzeuucxd3hhq
Improved Inapproximability of Rainbow Coloring
[article]
2018
arXiv
pre-print
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Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem. ...
We prove that given a rainbow (k - 2√(k))-colorable k-uniform hypergraph, it is NP-hard to find a normal 2-coloring. ...
We would also like to thank Florian Frick for bringing [LZ07] , [ACC + 18] , and the restrictions for Sarkaria's theorem to our notice. ...
arXiv:1810.02784v3
fatcat:dh6lpftmarepng4kiv26m646xe
Beyond the Borsuk-Ulam theorem: The topological Tverberg story
[article]
2017
arXiv
pre-print
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Then we introduce the "constraint method," which applied to suitable "unavoidable complexes" yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the "colored ...
with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." ...
We want to express our gratitude to Peter Landweber for his continuous help and support in improving this manuscript.
The Beginning 2.1. Radon's theorem. ...
arXiv:1605.07321v3
fatcat:tol2hry5s5fuxggejpunylr3ay
On Gromov's Method of Selecting Heavily Covered Points
[article]
2011
arXiv
pre-print
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These bounds yield a minor improvement over Gromov's lower bounds on c_d for large d, but they also show that the room for further improvement through the profiles alone is quite small. ...
We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for c_d. ...
If better lower bounds on ϕ 2 or ϕ 3 could be proved in suitable ranges of α, which would improve the lower bound (9), we would automatically get a further (slight) improvement from the proof below; in ...
arXiv:1102.3515v1
fatcat:3rpjf34bpvfqpjiagu7myonjke
On Gromov's Method of Selecting Heavily Covered Points
2014
Discrete & Computational Geometry
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These bounds yield a minor improvement over Gromov's lower bounds on c d for large d, but they also show that the room for further improvement through the cofilling profiles alone is quite small. ...
Thus, given a subset f ⊆ V , we will also sometimes refer to f as a face of n−1 , and the dimension of a face is defined as dim f := | f | − 1. ...
If better lower bounds on ϕ 2 or ϕ 3 could be proved in suitable ranges of α, which would improve the lower bound (9), we would automatically get a further (slight) improvement from the proof below; in ...
doi:10.1007/s00454-014-9584-7
fatcat:idtzopejuvhvxnzwdqwlrdqtcm
Inscribed Tverberg-Type Partitions for Orbit Polytopes
[article]
2022
arXiv
pre-print
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At the other extreme, one has polytopal partitions for d-polytopes on r vertices with isometry group equal to G whenever G is the isometry group of a vertex–transitive d-polytope. ...
As with Tverberg's theorem, the number of points is optimal for this. ...
Acknowledgements The authors are grateful to the anonymous referee, whose many thoughtful suggestions improved the clarity and presentation of the manuscript. ...
arXiv:2110.09322v3
fatcat:xw2mr5yuxneulp7gs5wotdlcty
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