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Kleitman and combinatorics

2002
*
Discrete Mathematics
*

Daniel Kleitman's many research contributions are surveyed, with emphasis on extremal hypergraph theory, asymptotic enumeration,

doi:10.1016/s0012-365x(02)00596-4
fatcat:ldvx5rh6gnc3lpwbzszc43d6hu
*and*discrete geometry. ... Acknowledgements I would like to thank Noga Alon, Navin Goyal, Curtis Greene, Je Kahn, Gyula Katona,*and*especially Doug West, for reading*a*previous version of this manuscript*and*for their many helpful ... comments*and*suggestions. ...##
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Sperner's Theorem and a Problem of Erdős, Katona and Kleitman

2014
*
Combinatorics, probability & computing
*

*A*central result in extremal set theory is the

*celebrated*theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing

*a*2-chain,F1⊂F2. ... Erdős

*and*Katona, followed by

*Kleitman*, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.In 1966,

*Kleitman*resolved this question for 2-chains, showing ... We hope that further work of this nature will lead to many interesting results

*and*

*a*greater understanding of classical theorems in extremal

*combinatorics*. ...

##
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Sperner's Theorem and a Problem of Erdos-Katona-Kleitman
[article]

2013
*
arXiv
*
pre-print

*A*central result in extremal set theory is the

*celebrated*theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing

*a*2-chain. ... Erdos

*and*Katona, followed by

*Kleitman*, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds. ... We hope that further work of this nature will lead to many interesting results

*and*

*a*greater understanding of classical theorems in extremal

*combinatorics*. ...

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Probabilistic Methods in Combinatorics
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

Let

doi:10.1007/978-3-0348-9078-6_132
fatcat:ok2d6v5smveq7osxkhc34hfcwy
*A*denote connectedness. In their most*celebrated*result Erdös*and*Rényi showed that if p = p(n) = ^ + ^ then Pr[*A*] -> exp(-e~c). We give [2], [6] as general references for these topics. ... Formally G(n,p) is*a*probability space whose points are graphs on*a*fixed labelled set of n vertices*and*where every pair of vertices is adjacent with independent probability p. ...*A*classic result of*Kleitman*[14] gives that some two Xi^X2 of these must differ in at least 2yn coordinates. Then X = (Xi ~ X2)/2 gives the desired partial coloring. ...##
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Refuting conjectures in extremal combinatorics via linear programming
[article]

2019
*
arXiv
*
pre-print

We apply simple linear programming methods

arXiv:1903.05495v1
fatcat:iuegd3t2krhebhypkvtz7dqy7i
*and*an LP solver to refute*a*number of open conjectures in extremal*combinatorics*. ... In the present manuscript we argue that the use of linear programming*and*LP solvers is such*a*method in extremal*combinatorics*. ... Theorem 3. 19 ( 19*Kleitman*[29]). Let s ≥ 2 be an integer*and*F ⊂ 2 [n]*a*family without s pairwise disjoint members. ...##
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Open problems in additive combinatorics
[chapter]

2007
*
CRM Proceedings and Lecture notes AMS
*

*A*brief historical introduction to the subject of additive

*combinatorics*

*and*

*a*list of challenging open problems, most of which are contributed by the leading experts in the area, are presented. ... Acknowledgement Some of the problems presented in this paper originate from the list, compiled by the present authors as

*a*follow-up to the Workshop on Recent Trends in Additive

*Combinatorics*, organized ... We are grateful to these institutions for bringing together

*a*large number of distinguished mathematicians, which ultimately allowed us to write this paper. ...

##
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Counting independent sets in graphs

2015
*
European journal of combinatorics (Print)
*

This method was first employed more than three decades ago by

doi:10.1016/j.ejc.2015.02.005
fatcat:ghgnxpmjszdp7cxmpj6ndg4ghm
*Kleitman**and*Winston*and*has subsequently been used numerous times by many researchers in various contexts. ... In particular, we derive bounds on the number of independent sets in regular graphs, sum-free subsets of {1, . . . , n},*and*C4-free graphs*and*give*a*short proof of an analogue of Roth's theorem on 3- ...*and*the*Kleitman*-Winston method*and*its applications over the past several years. ...##
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On the extremal combinatorics of the hamming space

1995
*
Journal of combinatorial theory. Series A
*

The first bounds on D(n) are those of

doi:10.1016/0097-3165(95)90019-5
fatcat:zrptyenaarfztngkftarv3sg6q
*Kleitman**and*Spencer [27] . ... ABOUT LANGUAGE--AN APOLOGY*Combinatorics*is cute. It speaks about graphs*and*hypergraphs, things you can draw*and*see. ...##
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Two-Part Set Systems

2012
*
Electronic Journal of Combinatorics
*

The two part Sperner theorem of Katona

doi:10.37236/2067
fatcat:g2zdsvkzujettehemcha6cp344
*and**Kleitman*states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$,*and*$\mathcal{F}$ is*a*family of subsets of $X$ such that no two sets $*A*, B \ ... \in\mathcal{F}$*and*$B \in \mathcal{G}$, then $*A*\not\subset B$*and*$B \not\subset*A*$. ... The*celebrated*theorems of Erdős, Ko, Rado [4]*and*of Sperner [13] determine the largest size that*a*uniform intersecting set system*and*Sperner system can have. ...##
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Book Review: Combinatorics with emphasis on the theory of graphs

1979
*
Bulletin of the American Mathematical Society
*

Its publication is

doi:10.1090/s0273-0979-1979-14606-8
fatcat:7tcgkkkjejcrfanpovhvwbhsnu
*a*notable event which affords the reviewer an opportunity to clarify his own ideas*and*to record his impressions of the present state of*combinatorics*. ... For one thing, combinatorial methods (as distinct from*combinatorics*as*a*subject) have naturally always constituted*a*vital ingredient of mathematical reasoning. ... the attention of*a*large number of mathematicians (among them Erdös, Katona,*Kleitman*,*and*Lovâsz). ...##
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Most Probably Intersecting Families of Subsets

2012
*
Combinatorics, probability & computing
*

Let F be

doi:10.1017/s0963548311000587
fatcat:lpvfpsthtrbo3fateytvc2mtri
*a*family of an n-element set. It is called intersecting if every pair of its members have*a*non-disjoint intersection. ... The new family is intersecting with*a*certain probability. We try to maximize this * ... Erdős, Dániel Gerbner, Balázs Keszegh,Ákos Kisvölcsey, Nathan Lemons, Dezső Miklós, Balázs Patkós, Attila Sali*and*Casey Tompkins. Gerbner also suggested*a*shorter proof. ...##
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Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New
[article]

2021
*
arXiv
*
pre-print

As well as being natural problems in their own right, intersection problems have connections with many other parts of

arXiv:2107.06371v8
fatcat:rcpfqcj3ijejxbs3xcqcn4rp7u
*Combinatorics**and*with Theoretical Computer Science (*and*indeed with many other parts ... The study of intersection problems in Extremal*Combinatorics*dates back perhaps to 1938, when Paul Erdős, Chao Ko*and*Richard Rado proved the (first) 'Erdős-Ko-Rado theorem' on the maximum possible size ... We thank an anonymous reviewer,*and*the editors of the Proceedings of the 29th BCC, for their careful reading of the paper,*and*for their helpful comments*and*suggestions, which we have incorporated. ...##
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The Dual BKR Inequality and Rudich's Conjecture

2010
*
Combinatorics, probability & computing
*

, then there exists

doi:10.1017/s0963548310000465
fatcat:zmswjt6hlvh3jogotwvmb7eogq
*a*term t ∈ T such that at least*a*δ-fraction of assignments satisfy some term of T sharing*a*variable with t [7]. ... (*A*key part of the proof is*a*correlation-like inequality on events in*a*finite product probability space that is in some sense dual to Reimer's inequality [10] a.k.a. the BKR inequality [4] or the van ... Theorem 1.6 (Harris-*Kleitman*Inequality [5, 9] ) For any finite product probability space (Ω, µ) with Ω = {0, 1} n ,*and**A*, B ⊆ Ω increasing, µ(*A*∩ B) ≥ µ(*A*)µ(B). ...##
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Combinatorics in the exterior algebra and the Bollobás Two Families Theorem
[article]

2020
*
arXiv
*
pre-print

We investigate the combinatorial structure of subspaces of the exterior algebra of

arXiv:1907.06019v2
fatcat:nqebxjdy5bf7vkditojo67y3u4
*a*finite-dimensional real vector space, working in parallel with the extremal*combinatorics*of hypergraphs. ... We also verify*a*recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins,*and*Xiao on pairs of set systems satisfying both an intersection*and**a*cross-intersection condition. ... Introduction For several decades there have been useful links between exterior algebra*and**combinatorics*. ...##
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Old and new applications of Katona's circle

2021
*
European journal of combinatorics (Print)
*

Concluding remarks

doi:10.1016/j.ejc.2021.103339
fatcat:6rkatgeyybfhhal4jz6zekqoum
*A*couple of months ago I decided to write*a*survey paper to*celebrate*the eightieth birthday of Gyula Katona, my ex-teacher*and*one of my best friends. ... The case s = 2 was proved by Greene, Katona*and**Kleitman*[28] . The proof is based on Corollary 4.6*and*the following operation discovered by Sperner. ...
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