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Nearly-linear monotone paths in edge-ordered graphs. (English) Zbl 1447.05113
Summary: How long is a monotone path one can always find in any edge-ordering of the complete graph \(K_n\)? This appealing question was first asked by V. Chvátal and J. Komlós [Can. Math. Bull. 14, 151–157 (1971; Zbl 0214.23503)], 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 \(n^{1-o(1)}\).
MSC:
05C38 Paths and cycles
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