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Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 588–612 | Cite as

Two extremal problems related to orders

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Abstract

We consider two extremal problems related to total orders on all subsets of \({\mathbb N}\). The first one is to maximize the Lagrangian of hypergraphs among all hypergraphs with m edges for a given positive integer m. In 1980’s, Frankl and Füredi conjectured that for a given positive integer m, the r-uniform hypergraph with m edges formed by taking the first m r-subsets of \({\mathbb N}\) in the colex order has the largest Lagrangian among all r-uniform hypergraphs with m edges. We provide some partial results for 4-uniform hypergraphs to this conjecture. The second one is for a given positive integer m, how to minimize the cardinality of the union closure families generated by edge sets of the r-uniform hypergraphs with m edges. Leck, Roberts and Simpson conjectured that the union closure family generated by the first m r-subsets of \({\mathbb N}\) in order U has the minimum cardinality among all the union closure families generated by edge sets of the r-uniform hypergraphs with m edges. They showed that the conjecture is true for graphs. We show that a similar result holds for non-uniform hypergraphs whose edges contain 1 or 2 vertices.

Keywords

Colex order Order U Union-closure Lagrangian of hypergraphs 

Notes

Acknowledgements

We thank the reviewers for reading the manuscript carefully, checking the details and giving insightful comments to help improve the manuscript.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Institute of MathematicsHunan UniversityChangshaPeople’s Republic of China

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