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Kleitman and combinatorics

Michael Saks
2002 Discrete Mathematics  
Daniel Kleitman's many research contributions are surveyed, with emphasis on extremal hypergraph theory, asymptotic enumeration, 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.  ... 
doi:10.1016/s0012-365x(02)00596-4 fatcat:ldvx5rh6gnc3lpwbzszc43d6hu

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.  ... 
doi:10.1017/s0963548314000273 fatcat:jrizeips55hrfphaz7eeao3y7i

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

Shagnik Das, Wenying Gan, Benny Sudakov
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.  ... 
arXiv:1302.5210v3 fatcat:avi66qr5zbdthh73fh7xjl26x4

Probabilistic Methods in Combinatorics [chapter]

Joel Spencer
1995 Proceedings of the International Congress of Mathematicians  
Let 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.  ... 
doi:10.1007/978-3-0348-9078-6_132 fatcat:ok2d6v5smveq7osxkhc34hfcwy

Refuting conjectures in extremal combinatorics via linear programming [article]

Adam Zsolt Wagner
2019 arXiv   pre-print
We apply simple linear programming methods 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.  ... 
arXiv:1903.05495v1 fatcat:iuegd3t2krhebhypkvtz7dqy7i

Open problems in additive combinatorics [chapter]

Ernest Croot, Vsevolod Lev
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.  ... 
doi:10.1090/crmp/043/10 fatcat:llqfcrpr2ja2baryr75c3o357u

Counting independent sets in graphs

Wojciech Samotij
2015 European journal of combinatorics (Print)  
This method was first employed more than three decades ago by 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.  ... 
doi:10.1016/j.ejc.2015.02.005 fatcat:ghgnxpmjszdp7cxmpj6ndg4ghm

On the extremal combinatorics of the hamming space

János Körner
1995 Journal of combinatorial theory. Series A  
The first bounds on D(n) are those of Kleitman and Spencer [27] .  ...  ABOUT LANGUAGE--AN APOLOGY Combinatorics is cute. It speaks about graphs and hypergraphs, things you can draw and see.  ... 
doi:10.1016/0097-3165(95)90019-5 fatcat:zrptyenaarfztngkftarv3sg6q

Two-Part Set Systems

Péter L. Erdős, Dániel Gerbner, Nathan Lemons, Dhruv Mubayi, Cory Palmer, Balázs Patkós
2012 Electronic Journal of Combinatorics  
The two part Sperner theorem of Katona 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.  ... 
doi:10.37236/2067 fatcat:g2zdsvkzujettehemcha6cp344

Book Review: Combinatorics with emphasis on the theory of graphs

L. Mirsky
1979 Bulletin of the American Mathematical Society  
Its publication is 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).  ... 
doi:10.1090/s0273-0979-1979-14606-8 fatcat:7tcgkkkjejcrfanpovhvwbhsnu

Most Probably Intersecting Families of Subsets

2012 Combinatorics, probability & computing  
Let F be 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.  ... 
doi:10.1017/s0963548311000587 fatcat:lpvfpsthtrbo3fateytvc2mtri

Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New [article]

David Ellis
2021 arXiv   pre-print
As well as being natural problems in their own right, intersection problems have connections with many other parts of 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.  ... 
arXiv:2107.06371v8 fatcat:rcpfqcj3ijejxbs3xcqcn4rp7u

The Dual BKR Inequality and Rudich's Conjecture

2010 Combinatorics, probability & computing  
, then there exists 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).  ... 
doi:10.1017/s0963548310000465 fatcat:zmswjt6hlvh3jogotwvmb7eogq

Combinatorics in the exterior algebra and the Bollobás Two Families Theorem [article]

Alex Scott, Elizabeth Wilmer
2020 arXiv   pre-print
We investigate the combinatorial structure of subspaces of the exterior algebra of 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.  ... 
arXiv:1907.06019v2 fatcat:nqebxjdy5bf7vkditojo67y3u4

Old and new applications of Katona's circle

Peter Frankl
2021 European journal of combinatorics (Print)  
Concluding remarks 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.  ... 
doi:10.1016/j.ejc.2021.103339 fatcat:6rkatgeyybfhhal4jz6zekqoum
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