Two extremal problems related to orders

Biao Wu () and Yuejian Peng ()
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Biao Wu: Hunan Normal University
Yuejian Peng: Hunan University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 2, No 19, 588-612

Abstract: Abstract We consider two extremal problems related to total orders on all subsets of $${\mathbb N}$$ 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 $${\mathbb N}$$ 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 $${\mathbb N}$$ 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 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0196-z

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