Abstract
A well-known conjecture of Frankl and Füredi from 1989 states that an initial segment of the colexicographic order has the largest Lagrangian of any r-uniform hypergraph with m hyperedges. We show that this is true when r = 3. We also give a new proof of a related conjecture of Nikiforov for large t and a counterexample to an old conjecture of Ahlswede and Katona.
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Acknowledgements
This research was partially completed while the third author was visiting ETH Zurich. The third author would like to thank Benny Sudakov and the London Mathematical Society for making this visit possible. We would like to thank Rob Morris and Imre Leader for their helpful comments and advice.
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The second author was supported by Dr. Max Rössler, the Walter Haefner Foundation, the ETH Zurich Foundation and and Murray Edwards College, Cambridge.
The third author is supported by a research fellowship from Sidney Sussex College, Cambridge.
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Gruslys, V., Letzter, S. & Morrison, N. Lagrangians of hypergraphs II: When colex is best. Isr. J. Math. 242, 637–662 (2021). https://doi.org/10.1007/s11856-021-2132-2
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DOI: https://doi.org/10.1007/s11856-021-2132-2