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Nearly-linear monotone paths in edge-ordered graphs

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Abstract

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, 8092, Zurich, Switzerland

    Matija Bucić, Benny Sudakov & Adam Zsolt Wagner

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Matthew Kwan

  3. Department of Economics, Mathematics, and Statistics, Birkbeck University of London, London, WC1E 7HX, UK

    Alexey Pokrovskiy

  4. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

    Tuan Tran

Authors
  1. Matija Bucić
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  2. Matthew Kwan
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  3. Alexey Pokrovskiy
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  4. Benny Sudakov
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  5. Tuan Tran
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  6. Adam Zsolt Wagner
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Corresponding author

Correspondence to Benny Sudakov.

Additional information

Research supported in part by SNSF project 178493.

Research supported in part by SNSF grant 200021-175573.

Research supported by the Humboldt Research Foundation.

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Bucić, M., Kwan, M., Pokrovskiy, A. et al. Nearly-linear monotone paths in edge-ordered graphs. Isr. J. Math. 238, 663–685 (2020). https://doi.org/10.1007/s11856-020-2035-7

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  • DOI: https://doi.org/10.1007/s11856-020-2035-7

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