# Two extremal problems related to orders

- 125 Downloads

## 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.

## References

- Frankl P, Füredi Z (1989) Extremal problems whose solutions are the blow-ups of the small Witt-designs. J Comb Theory 52:129–147CrossRefMATHGoogle Scholar
- Frankl P, Rödl V (1984) Hypergraphs do not jump. Combinatorica 4:149–159MathSciNetCrossRefMATHGoogle Scholar
- He S, Li Z, Zhang S (2010) Approximation algorithms for homogenous polynomial optimization with quadratic constraints. Math Program Ser B 125:353–383CrossRefMATHGoogle Scholar
- Keevash P (2011) Hypergrah Turán problems, surveys in combinatorics. Cambridge University Press, Cambridge, pp 83–140MATHGoogle Scholar
- Kim H (2011) The union closure method for testing a fixed sequence of families of hypotheses. Biometrika 98:391–401MathSciNetCrossRefMATHGoogle Scholar
- Leck U, Roberts I, Simpson J (2012) Minimizing the weight of the union-closure of families of two-sets. Australas J Combin 52:67–73MathSciNetMATHGoogle Scholar
- Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of Turán. Can J Math 17:533–540CrossRefMATHGoogle Scholar
- Sidorenko AF (1987) Solution of a problem of Bollobas on 4-graphs. Mat Zametki 41:433–455MathSciNetGoogle Scholar
- Sun YP, Tang QS, Zhao C, Peng Y (2014) On the largest graph-Lagrangian of 3-graphs with fixed number of edges. J Optim Theory Appl 163(1):57–79MathSciNetCrossRefMATHGoogle Scholar
- Tang QS, Peng H, Wang C, Peng Y (2016) On Frankl and Füredi’s conjecture for 3-uniform hypergraphs. Acta Math Appl Sin English Ser 32(1):95–112MathSciNetCrossRefMATHGoogle Scholar
- Tang QS, Peng Y, Zhang XD, Zhao C (2014) Some results on Lagrangians of hypergraphs. Discrete Appl Math 166:222–238MathSciNetCrossRefMATHGoogle Scholar
- Tang QS, Peng Y, Zhang XD, Zhao C (2016) Connection between the clique number and the Lagrangian of 3-uniform hypergraphs. Optim Lett 10(4):685–697MathSciNetCrossRefMATHGoogle Scholar
- Talbot J (2002) Lagrangians of hypergraphs. Combin Probab Comput 11:199–216MathSciNetCrossRefMATHGoogle Scholar
- Turán P (1941) On an extremal problem in graph theory (in Hungarian). Mat Fiz Lapok 48:436–452MathSciNetGoogle Scholar