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Extremals of functions on graphs with applications to graphs and hypergraphs

Vera T Sós, E.G Straus
1982 Journal of combinatorial theory. Series B (Print)  
The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P.  ...  The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges.  ...  In Section 4 we give applications to graphs and hypergraphs. THE MAIN THEOREM We first state a general combinatorial inequality and then give a few more concrete versions. THEOREM 2.1.  ... 
doi:10.1016/0095-8956(82)90002-8 fatcat:7amsi7srdfb2hjk2uxaiyqdiam

Improved bounds on the extremal function of hypergraphs [article]

William Zhang
2018 arXiv   pre-print
We use the equivalence to prove several new bounds on the extremal functions of graphs.  ...  We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite graphs and the asymptotics of the matrix extremal function.  ...  Acknowledgements The author would like to thank Dr. Jesse Geneson for his valuable advice and guidance through the duration of this project.  ... 
arXiv:1807.02411v1 fatcat:igfsh4pzg5g2zj27hvwvmqgx2u

Models and Methods for Sparse (Hyper)Network Science in Business, Industry, and Government

Sinan G Aksoy, Aric Hagberg, Cliff A Joslyn, Bill Kay, Emilie Purvine, Stephen J Young
2022 Notices of the American Mathematical Society  
model multi-way relations in hypergraphs, rather than only pairwise interactions in graphs; and (2) challenges posed by modeling networks with extreme sparsity.  ...  The authors are hosting an AMS sponsored Mathematics Research Community (MRC) focusing on two themes that have garnered intense attention in network models of complex relational data: (1) how to faithfully  ...  In contrast with graphs, these width-based partitions induce non-trivial filtrations on a set of hypergraph walks.  ... 
doi:10.1090/noti2424 fatcat:hnu5w6rojraudjkchhm4k5cdrq

Hypergraph products for structural mechanics

A. Kaveh, B. Alinejad
2015 Advances in Engineering Software  
Variable adjacency functions on hypergraphs result in generation of dynamic graph products which are applied to dynamic systems.  ...  In this paper Cartesian, direct and strong Cartesian product of hypergraphs are investigated. A new concept named the adjacency function is defined on hypergraphs.  ...  Hypergraphs are generalized forms of graphs and many applications of graphs can be extended to hypergraphs because of their generality [6, 7] .  ... 
doi:10.1016/j.advengsoft.2014.09.017 fatcat:cut6sfzxuvg4ncdx7csvt3mlwy

An extension of Motzkin-Straus Theorem to non-uniform hypergraphs and its applications [article]

Yuejian Peng, Hao Peng, Qingsong Tang, Cheng Zhao
2013 arXiv   pre-print
The connection of Lagrangians and Turán densities can be also used to prove the fundamental theorem of Erdös-Stone-Simonovits on Turán densities of graphs.  ...  In this paper, we attempt to explore the applications of Lagrangian method in determining Turán densities of non-uniform hypergraphs.  ...  This research is partially supported by National Natural Science Foundation of China (No. 11271116).  ... 
arXiv:1312.4135v1 fatcat:oxud775onvdx5nq5rwf42g3vfq

Regular slices for hypergraphs

Peter Allen, Julia Böttcher, Oliver Cooley, Richard Mycroft
2015 Electronic Notes in Discrete Mathematics  
We believe that many arguments in extremal hypergraph theory are made considerably simpler by using this lemma rather than existing forms of the Strong Hypergraph Regularity Lemma, and advocate its use  ...  We present a 'Regular Slice Lemma' which, given a k-graph G, returns a regular (k − 1)-complex J with respect to which G has useful regularity properties.  ...  The difficulty with these tools is their technical intricacy, which leads to significant additional complexity in applications of the regularity method in extremal hypergraph theory.  ... 
doi:10.1016/j.endm.2015.06.093 fatcat:admnbs56sneafhikqnf2sdihbm

Extremal problems for ordered (hyper)graphs: applications of Davenport-Schinzel sequences [article]

Martin Klazar
2003 arXiv   pre-print
We extend, by means of a more generally applicable theorem, the n.log n upper bound on the ordered graph extremal function of F=(1,3, 1,5, 2,3, 2,4) due to Z. Furedi to the n.(log n)^2.  ...  We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions.  ...  Introduction and motivation In this article we shall investigate extremal problems on graphs and hypergraphs of the following type.  ... 
arXiv:math/0305037v1 fatcat:s3qfh76o25dohdbpu54hiybghy

Page 1722 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
G. 84e:05065 Extremals of functions on graphs with applications to graphs and hypergraphs. J. Combin. Theory Ser. B 32 (1982), no. 3, 246-257. Authors’ summary: “The method used in an article by T.  ...  The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P.  ... 

Extremal problems for ordered (hyper)graphs: applications of Davenport–Schinzel sequences

Martin Klazar
2004 European journal of combinatorics (Print)  
We extend, using a more generally applicable theorem, the n log n upper bound on sizes of ({1, 3}, {1, 5}, {2, 3}, {2, 4})-free ordered graphs with n vertices, due to Füredi, to the n(log n) 2 (log log  ...  We introduce a containment relation of hypergraphs which respects linear orderings of vertices, and we investigate associated extremal functions.  ...  Acknowledgements ITI is supported by the project LN00A056 of the Czech Ministry of Education. The author is grateful to an anonymous referee for several stylistic suggestions.  ... 
doi:10.1016/j.ejc.2003.05.001 fatcat:zn73ftlbrre43pysnpawajd5b4

Ramsey–Turán theory

Miklós Simonovits, Vera T. Sós
2001 Discrete Mathematics  
Motivated by an observation of Paul Erdős, it was Turà an who started the systematic investigation of the applications of extremal graph theory in geometry and analysis.  ...  For every L 1 ; : : : ; L r there exists a function f(n) = o(n) for which RT(n; L 1 ; : : : ; L r ; f(n)) = #(L 1 ; : : : ; L r )n 2 + o(n 2 ): See also Deÿnition 52 and Problem 9 on threshold functions  ...  Acknowledgements We thank Stephan Brandt and Zoltà an F uredi for their valuable remarks.  ... 
doi:10.1016/s0012-365x(00)00214-4 fatcat:2nfhfsiozna5lkc2flkpubbkva

An Efficient and Scalable Distributed Hypergraph Processing System

Shugo Fujimura, Shigeyuki Sato, Kenjiro Taura
2021 Journal of Information Processing  
Our system outperformed the state-of-the-art shared-memory hypergraph processing system Hygra from several folds to tens of folds on a single-node computer.  ...  The recent increase in the amount of graph data has drawn our attention to distributed graph processing systems scalable to large-scale inputs.  ...  We therefore consider the design and implementation of Gemini to be extremely promising for hypergraph processing dealt with by Hygra.  ... 
doi:10.2197/ipsjjip.29.812 fatcat:hpuxzoriovajtjerc3onzczei4

Extremal Hypergraphs and Combinatorial Geometry [chapter]

Zoltán Füredi
1995 Proceedings of the International Congress of Mathematicians  
Here we overview some of the methods and results of extremal graph and hypergraph theory. A few geometric applications are also given.  ...  Wc overview sonic of the methods used in extremal graph and hypergraph theory and illustrate them by Turan-type problems.  ...  The best upper bound is due to Clarkson et al. [13] , f^3\n) < 0(n 3 / 2 ß(n)), where ß(n) is an extremely slowly growing function related to the inverse of Ackermann's function.  ... 
doi:10.1007/978-3-0348-9078-6_129 fatcat:oo5262xz2ncsdn5vndwklwufri

Recent Developments in Extremal Combinatorics: Ramsey and Turán Type Problems

Benny Sudakov
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades.  ...  In this paper we discuss some recent advances in this subject, focusing on two topics which played an important role in the development of extremal combinatorics: Ramsey and Turán type questions for graphs  ...  Acknowledgments I would like to thank Conlon, Fox, Keevash, Krivelevich and Markman for carefully reading this manuscript and many valuable comments, which greatly improved the presentation.  ... 
doi:10.1142/9789814324359_0159 fatcat:6dzzmc7s3bbufe36nr3k7mh5vi

The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphs [article]

Gabriela Araujo-Pardo, Juan Carlos Díaz-Patiño, Luis Montejano, Deborah Oliveros
2014 arXiv   pre-print
large and p ≥ q ≥ 3 by proposing it as a problem of extremal graph theory.  ...  In this paper we have found a link between the fractional chromatic number of Kneser hypergraphs and the (p,q)-extremal problem and also solve the (p,q)-extremal problem for graphs if n is sufficiently  ...  Acknowledgements The authors wish to acknowledge support from CONACyT under projects 166306, 178395 and support from PAPIIT-UNAM under projects IN101912, IN112614.  ... 
arXiv:1408.3197v1 fatcat:fyuh5jtmpfg4rd442wwrb6g3oq

The method of hypergraph containers [article]

József Balogh, Robert Morris, Wojciech Samotij
2018 arXiv   pre-print
We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics  ...  In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures.  ...  graphs and on extremal and Ramsey properties of random discrete structures inspired and influenced our investigations that led to the hypergraph container theorems.  ... 
arXiv:1801.04584v1 fatcat:2i524zhma5gbloc7ovoiq2znmi
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