Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph K n ? 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 o...

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Published inIsrael journal of mathematics Vol. 238; no. 2; pp. 663 - 685
Main Authors Bucić, Matija, Kwan, Matthew, Pokrovskiy, Alexey, Sudakov, Benny, Tran, Tuan, Wagner, Adam Zsolt
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.07.2020
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Lower bounds
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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ISSN0021-2172
1565-8511
DOI10.1007/s11856-020-2035-7

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Summary:How long a monotone path can one always find in any edge-ordering of the complete graph K n ? 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 n 2/3- 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 n 1-o(1) .
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ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-020-2035-7