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Intersection theorems and a lemma of Kleitman

Ian Anderson
1976 Discrete Mathematics  
Anderson / 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!  ... 
doi:10.1016/0012-365x(76)90097-2 fatcat:aitmlg5o2nc3tceeklmwvb4wxq

The (2,2) and (4,3) properties in families of fat sets in the plane [article]

Shiliang Gao, Shira Zerbib
2017 arXiv   pre-print
This extends results by Danzer 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.  ... 
arXiv:1711.05308v1 fatcat:ct35wytvyregjg6j3qul42pb7u

Notes on Chvátal's conjecture

Yi Wang
2002 Discrete Mathematics  
Using Kleitman's 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.  ... 
doi:10.1016/s0012-365x(01)00317-x fatcat:awinsz5gfnc3jctfuheioqenou

Improved bounds on the Hadwiger–Debrunner numbers

Chaya Keller, Shakhar Smorodinsky, Gábor Tardos
2018 Israel Journal of Mathematics  
In 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.  ... 
doi:10.1007/s11856-018-1685-1 fatcat:7a3si3yk5zclfhrmqglp4kn2we

On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Chaya Keller, Shakhar Smorodinsky, Gábor Tardos
2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms  
In 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.  ... 
doi:10.1137/1.9781611974782.148 dblp:conf/soda/KellerST17 fatcat:bhcytgudnfblzeibvhlmcvotp4

Improved bounds on the Hadwiger-Debrunner numbers [article]

Chaya Keller and Shakhar Smorodinsky and Gabor Tardos
2016 arXiv   pre-print
In 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.  ... 
arXiv:1512.04026v3 fatcat:hgdizui2yraepegahictnbijwm

On Piercing Numbers of Families Satisfying the (p,q)_r Property [article]

Chaya Keller, Shakhar Smorodinsky
2017 arXiv   pre-print
Almost tight upper bounds for HD_d(p,q) for 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.  ... 
arXiv:1703.06338v1 fatcat:qlk75i73sfbmdebmbeuzisa3bu

About an Erdős–Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

Amanda Montejano, Luis Montejano, Edgardo Roldán-Pensado, Pablo Soberón
2015 Discrete & Computational Geometry  
In this paper, we study the number 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) .  ... 
doi:10.1007/s00454-015-9664-3 fatcat:26riqgtq7fdwdb2n4zldho6rc4

Further Consequences of the Colorful Helly Hypothesis

Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, Natan Rubin
2019 Discrete & Computational Geometry  
The Colorful Helly 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.  ... 
doi:10.1007/s00454-019-00085-y fatcat:e255gxemvrftdai5a44sz4icby

An Improved Kalai--Kleitman Bound for the Diameter of a Polyhedron

Michael J. Todd
2014 SIAM Journal on Discrete Mathematics  
Kalai 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.  ... 
doi:10.1137/140962310 fatcat:2m7ait2tgrcgbnxpvvkol6mbm4

Short proof of two cases of Chvátal's conjecture [article]

Jorge Olarte, Francisco Santos, Jonathan Spreer
2018 arXiv   pre-print
In the same year 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]).  ... 
arXiv:1804.03646v2 fatcat:mjjfwidgibegvg3k2n6borq2le

About an Erdős-Grünbaum conjecture concerning piercing of non bounded convex sets [article]

Amanda Montejano, Luis Montejano, Edgardo Roldán-Pensado, Pablo Soberón
2014 arXiv   pre-print
In this paper, we study the number 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.  ... 
arXiv:1407.0642v2 fatcat:a6v6b4u3xfezxbuuvweo23nhdy

Equality in a result of Kleitman

Daniel McQuillan, R.Bruce Richter
1994 Journal of combinatorial theory. Series A  
An upset is 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.  ... 
doi:10.1016/0097-3165(94)90029-9 fatcat:dwd2nnabj5d2pbq67mgpxusjgm

Orthogonal Symmetric Chain Decompositions of Hypercubes [article]

Hunter Spink
2017 arXiv   pre-print
In 1979, Shearer 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.  ... 
arXiv:1706.08545v2 fatcat:yxj7czb2vzcu3fwgwclo5deh3u

The Edge-Diametric Theorem in Hamming Spaces

Christian Bey
2005 Electronic Notes in Discrete Mathematics  
The binary case was solved earlier by Ahlswede 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).  ... 
doi:10.1016/j.endm.2005.07.036 fatcat:yqclw27tonfdxjh32rzybwrvbi
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