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Intersection theorems and a lemma of Kleitman

1976
*
Discrete Mathematics
*

Anderson /

doi:10.1016/0012-365x(76)90097-2
fatcat:aitmlg5o2nc3tceeklmwvb4wxq
*Intersection*tkeunrms*and**a**lemma**of*KIeitman Further, if 9 is*a*set*of*divisors*of*m*and*if %'( CR) denotes the set*of*;! ... *'.*and*the*theorem*is proved.*A*conjectwe*of*IKatona Katona !_'I has prcjved that if*A*I, . . . .*A*. are subsets*of*S stch that iA, 1'3*A*! ...##
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The (2,2) and (4,3) properties in families of fat sets in the plane
[article]

2017
*
arXiv
*
pre-print

This extends results by Danzer

arXiv:1711.05308v1
fatcat:ct35wytvyregjg6j3qul42pb7u
*and*Karasev on the piercing numbers in*intersecting*families*of*disks in the plane, as well as*a*result by Kynčl*and*Tancer on the piercing numbers in families*of*units disks ...*A*family*of*sets satisfies the (p,q) property if among every p members*of*it some q*intersect*. ... We are grateful to the Department*of*Mathematics at the University*of*Michigan for supporting this project. ...##
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Notes on Chvátal's conjecture

2002
*
Discrete Mathematics
*

Using Kleitman's

doi:10.1016/s0012-365x(01)00317-x
fatcat:awinsz5gfnc3jctfuheioqenou
*lemma**and*results*of*Sch onheim*and*Miklà os it is shown that if w(D) = |D|=2, then every maximum-sized*intersecting*family in D contains all base elements*of*D. ... Then, the converse*of*this statement is conjectured*and*shown that this is equivalent to that*of*Chvà atal. ... Acknowledgements The author would like to thank the anonymous referees for their careful reading*and*valuable suggestions. ...##
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Improved bounds on the Hadwiger–Debrunner numbers

2018
*
Israel Journal of Mathematics
*

In

doi:10.1007/s11856-018-1685-1
fatcat:7a3si3yk5zclfhrmqglp4kn2we
*a*celebrated proof*of*the Hadwiger-Debrunner conjecture, Alon*and**Kleitman*proved that We present several improved bounds: (iii) For every ǫ > 0 there exists*a*p 0 = p 0 (ǫ) such that for every p ≥ ... Based on this, we introduce*a*polynomial time constant factor approximation algorithm for MAX-CLIQUE*of**intersection*graphs*of*convex sets satisfying this property. ... Acknowledgements We wish to thank Noga Alon*and*Andreas Holmsen for helpful comments. ...##
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On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

2017
*
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
*

In

doi:10.1137/1.9781611974782.148
dblp:conf/soda/KellerST17
fatcat:bhcytgudnfblzeibvhlmcvotp4
*a*celebrated proof*of*the Hadwiger-Debrunner conjecture, Alon*and**Kleitman*proved that This paper has two parts. In the first part we present several improved bounds on HD d (p, q). ... Based on this, we introduce*a*polynomial time constant factor approximation algorithm for MAX-CLIQUE*of**intersection*graphs*of*convex sets with sub-quadratic union complexity. ... Acknowledgements We wish to thank Noga Alon*and*Andreas Holmsen for helpful comments. ...##
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Improved bounds on the Hadwiger-Debrunner numbers
[article]

2016
*
arXiv
*
pre-print

In

arXiv:1512.04026v3
fatcat:hgdizui2yraepegahictnbijwm
*a*celebrated proof*of*the Hadwiger-Debrunner conjecture, Alon*and**Kleitman*proved that HD_d(p,q) exists for all p ≥ q ≥ d+1. Specifically, they prove that HD_d(p,d+1) is Õ(p^d^2+d). ... Based on this, we introduce*a*polynomial time constant factor approximation algorithm for MAX-CLIQUE*of**intersection*graphs*of*convex sets satisfying this property. ... Acknowledgements We wish to thank Noga Alon*and*Andreas Holmsen for helpful comments. ...##
###
On Piercing Numbers of Families Satisfying the (p,q)_r Property
[article]

2017
*
arXiv
*
pre-print

Almost tight upper bounds for HD_d(p,q) for

arXiv:1703.06338v1
fatcat:qlk75i73sfbmdebmbeuzisa3bu
*a*'sufficiently large' q were obtained recently using an enhancement*of*the celebrated Alon-*Kleitman**theorem*, but no sharp upper bounds for*a*general q are known ... ., 45(2) (2011), pp. 358-364], Montejano*and*Soberón defined*a*refinement*of*the (p,q) property: F satisfies the (p,q)_r property if among any p elements*of*F, at least r*of*the q-tuples*intersect*. ... First we state*a**lemma**of*[9] on which we base our argument. 3.1 The technique*of*[9]*and*an alternative proof*of*the Hadwiger-Debrunner*theorem**Lemma*3.1. ...##
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About an Erdős–Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

2015
*
Discrete & Computational Geometry
*

In this paper, we study the number

doi:10.1007/s00454-015-9664-3
fatcat:26riqgtq7fdwdb2n4zldho6rc4
*of*compact sets needed in an infinite family*of*convex sets with*a*local*intersection*structure to imply*a*bound on its piercing number, answering*a*conjecture*of*Erdős ...*and*Grünbaum. ... Figure 1 : 1 Simplexes S α*and*S β in R 2*and*R 3 .*Lemma*2. 3 . 3 The piercing number π(*A*) = ∞. Figure 2 : 2 The unbounded sets*of*F in R 2 .*Theorem*3. 1 ( 1 Alon,*Kleitman*1992) . ...##
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Further Consequences of the Colorful Helly Hypothesis

2019
*
Discrete & Computational Geometry
*

The Colorful Helly

doi:10.1007/s00454-019-00085-y
fatcat:e255gxemvrftdai5a44sz4icby
*Theorem**of*Lovász states that for any such colorful family F there is*a*color class F i ⊂ F, for 1 ≤ i ≤ d+1, whose sets have*a*non-empty*intersection*. ... We say that F satisfies the Colorful Helly Property if every rainbow selection*of*d + 1 sets, one set from each color class, has*a*non-empty common*intersection*. ... Acknowledgements The authors thank the anonymous SoCG referees for valuable comments which helped to improve the presentation*of*the paper. ...##
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An Improved Kalai--Kleitman Bound for the Diameter of a Polyhedron

2014
*
SIAM Journal on Discrete Mathematics
*

Kalai

doi:10.1137/140962310
fatcat:2m7ait2tgrcgbnxpvvkol6mbm4
*and**Kleitman*[6] established the bound log( )+2 for the diameter*of**a*-dimensional polyhedron with facets. Here we improve the bound slightly to ( − ) log( ) . ... Acknowledgement Thanks to Günter Ziegler, Francisco Santos,*and*the referees for several helpful comments on previous versions. ... Our proof*of*the improved bound uses the same*lemma*as employed by Kalai*and**Kleitman*, with*a*slightly tighter analysis*of*the inductive step*and*the consideration*of**a*number*of*low-dimensional cases. ...##
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Short proof of two cases of Chvátal's conjecture
[article]

2018
*
arXiv
*
pre-print

In the same year

arXiv:1804.03646v2
fatcat:mjjfwidgibegvg3k2n6borq2le
*Kleitman**and*Magnanti proved the conjecture when F is contained in the union*of*two stars,*and*Sterboul when rank(F)< 3. We give short self-contained proofs*of*these two statements. ... In 1974 Chvátal conjectured that no*intersecting*family F in*a*downset can be larger than the largest star. ... Once we know there are exactly six vertices, observe that at most half*of*the*Theorem*1.2 (*Kleitman**and*Magnanti [4,*Theorem*2]). ...##
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About an Erdős-Grünbaum conjecture concerning piercing of non bounded convex sets
[article]

2014
*
arXiv
*
pre-print

In this paper, we study the number

arXiv:1407.0642v2
fatcat:a6v6b4u3xfezxbuuvweo23nhdy
*of*compact sets needed in an infinite family*of*convex sets with*a*local*intersection*structure to imply*a*bound on its piercing number, answering*a*conjecture*of*Erdős ...*and*Grünbaum. ... then, by*Lemma*2.2, all the elements in*A*∪ B*intersect*. 2. If d + 1 ≤ i ≤ d + k then, by*Lemma*2.2, d*of*the elements in*A**and*all the elements*of*B*intersect*. ...##
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Equality in a result of Kleitman

1994
*
Journal of combinatorial theory. Series A
*

An upset is

doi:10.1016/0097-3165(94)90029-9
fatcat:dwd2nnabj5d2pbq67mgpxusjgm
*a*set q/*of*subset*of**a*finite set. S such that if U~ V*and*Ueq/, then V~ql.*A*downset 9 is defined analogously. In 1966,*Kleitman*(J. Combin. ... In this note, we show that*a*necessary*and*sufficient condition for equality to hold is: for every minimal element U*of*og*and*every maximal element D*of*9, U_ D. ... . | We now move on to two corollaries*of**Theorem*1; the inequality in the second is Kleitman's*Theorem*on the size*of*the union*of**intersecting*families I-3]. COROLLARY 1.1. ...##
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Orthogonal Symmetric Chain Decompositions of Hypercubes
[article]

2017
*
arXiv
*
pre-print

In 1979, Shearer

arXiv:1706.08545v2
fatcat:yxj7czb2vzcu3fwgwclo5deh3u
*and**Kleitman*conjectured that there exist n/2 +1 orthogonal chain decompositions*of*the hypercube Q_n,*and*constructed two orthogonal chain decompositions. ... We explicitly describe three such decompositions*of*Q_5*and*Q_7,*and*describe conditions which allow us to decompose products*of*hypercubes into k almost orthogonal symmetric chain decompositions given ... Proof*of**Theorems*3.1*and*3.3 We will prove*Theorem*3.3 through*a*series*of**lemmas*,*and*then prove*Theorem*3.1 as*a*consequence. ...##
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The Edge-Diametric Theorem in Hamming Spaces

2005
*
Electronic Notes in Discrete Mathematics
*

The binary case was solved earlier by Ahlswede

doi:10.1016/j.endm.2005.07.036
fatcat:yqclw27tonfdxjh32rzybwrvbi
*and*Khachatrian [*A*diametric*theorem*for edges, J. Combin. Theory Ser.*A*92(1) (2000) 1-16]. ... The maximum number*of*edges spanned by*a*subset*of*given diameter in*a*Hamming space with alphabet size at least three is determined. ... Equivalent versions*of*the above*theorems*(in terms*of**intersection*instead*of*diametry) were obtained by Katona [10] (*Theorem*1)*and*Frankl*and*Tokushige [9] (*Theorem*2). ...
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