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A Slight Improvement to the Colored Bárány's Theorem [article]

Zilin Jiang
2014 arXiv   pre-print
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

Zilin Jiang
unpublished
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]

Frédéric Meunier, Pauline Sarrabezolles
2016 arXiv   pre-print
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]

Andrey Borisovich Kupavskii
2015 arXiv   pre-print
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]

Boris Bukh, Alfredo Hubard
2017 arXiv   pre-print
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

Nabil H. Mustafa, Saurabh Ray
2016 Discrete Mathematics  
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

Nabil H. Mustafa, Saurabh Ray
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
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]

Megumi Asada, Florian Frick, Vivek Pisharody, Maxwell Polevy, David Stoner, Ling Hei Tsang, Zoe Wellner
2017 arXiv   pre-print
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]

Andrey Kupavskii
2018 arXiv   pre-print
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

Andrey Kupavskii
2018 Electronic Journal of Combinatorics  
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]

Per Austrin, Amey Bhangale, Aditya Potukuchi
2018 arXiv   pre-print
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]

Pavle V. M. Blagojević, Günter M. Ziegler
2017 arXiv   pre-print
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]

Jiří Matoušek, Uli Wagner
2011 arXiv   pre-print
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

Jiří Matoušek, Uli Wagner
2014 Discrete & Computational Geometry  
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]

Steven Simon, Tobias Timofeyev
2022 arXiv   pre-print
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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