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) .

中文翻译：

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边序图中的近线性单调路径

在完整图 K n 的任何边序中总能找到多长的单调路径？这个吸引人的问题于 1971 年由 Chvátal 和 Komlós 首次提出，此后引起了许多研究人员的关注，激发了各种相关问题。普遍的猜想是，人们总是可以找到线性长度的单调路径，但直到现在，最著名的下界是 n 2/3- o (1) 。在本文中，我们几乎缩小了这个差距，证明了完整图的任何边排序都包含长度为 n 1-o(1) 的单调路径。