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.
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Yuejian Peng: Partially supported by National Natural Science Foundation of China (No. 11671124).
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Wu, B., Peng, Y. Two extremal problems related to orders. J Comb Optim 35, 588–612 (2018). https://doi.org/10.1007/s10878-017-0196-z
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DOI: https://doi.org/10.1007/s10878-017-0196-z