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

2014
*
arXiv
*
pre-print

In this paper, we prove that there exists

arXiv:1405.2503v2
fatcat:4deh2fsutjgjvfwafvaxh4dsqy
*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. ...##
###
A Slight Improvement to the Colored Bárány's Theorem

unpublished

In this paper, we prove that there exists

fatcat:2xj2tm4zgngonnzocahferw2ha
*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. ...##
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Colorful linear programming, Nash equilibrium, and pivots
[article]

2016
*
arXiv
*
pre-print

In 1997, Bárány and Onn defined

arXiv:1409.3436v3
fatcat:ghi6me4jxzhbbemup263pwas7i
*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. ...##
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On random subgraphs of Kneser and Schrijver graphs
[article]

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. ...

##
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On a topological version of Pach's overlap theorem
[article]

2017
*
arXiv
*
pre-print

This

arXiv:1708.04350v1
fatcat:vuqg232sjbebpbljs2rzbb4dbe
*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. ...##
###
An optimal generalization of the Colorful Carathéodory theorem

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. ...

##
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A theorem of bárány revisited and extended

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. ...

##
###
Fair division and generalizations of Sperner- and KKM-type results
[article]

2017
*
arXiv
*
pre-print

Furthermore, we extend Sperner's lemma and

arXiv:1701.04955v1
fatcat:6m5zxxh4f5dknc4aef7yzkihni
*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*. ...##
###
Random Kneser graphs and hypergraphs
[article]

2018
*
arXiv
*
pre-print

This allows us

arXiv:1612.03868v3
fatcat:ghdrlmw4hfgwjklfwo4bben4hq
*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. ...##
###
Random Kneser Graphs and Hypergraphs

2018
*
Electronic Journal of Combinatorics
*

Moreover, they have

doi:10.37236/8005
fatcat:so6rixj5l5ba7lbzeuucxd3hhq
*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. ...##
###
Improved Inapproximability of Rainbow Coloring
[article]

2018
*
arXiv
*
pre-print

Theory. 1990) using topological methods and

arXiv:1810.02784v3
fatcat:dh6lpftmarepng4kiv26m646xe
*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. ...##
###
Beyond the Borsuk-Ulam theorem: The topological Tverberg story
[article]

2017
*
arXiv
*
pre-print

Then we introduce

arXiv:1605.07321v3
fatcat:tol2hry5s5fuxggejpunylr3ay
*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*. ...##
###
On Gromov's Method of Selecting Heavily Covered Points
[article]

2011
*
arXiv
*
pre-print

These bounds yield

arXiv:1102.3515v1
fatcat:3rpjf34bpvfqpjiagu7myonjke
*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 ...##
###
On Gromov's Method of Selecting Heavily Covered Points

2014
*
Discrete & Computational Geometry
*

These bounds yield

doi:10.1007/s00454-014-9584-7
fatcat:idtzopejuvhvxnzwdqwlrdqtcm
*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 ...##
###
Inscribed Tverberg-Type Partitions for Orbit Polytopes
[article]

2022
*
arXiv
*
pre-print

At

arXiv:2110.09322v3
fatcat:xw2mr5yuxneulp7gs5wotdlcty
*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. ...
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