Nearly-linear monotone paths in edge-ordered graphs

TL;DR

This paper proves that in any edge-ordering of a complete graph, there always exists a monotone path whose length is nearly linear in the number of vertices, significantly improving previous bounds.

Contribution

It establishes a nearly-linear lower bound on the length of monotone paths in edge-ordered complete graphs, nearly closing the long-standing conjecture.

Findings

Monotone paths of length n^{1-o(1)} always exist in edge-ordered K_n.

Improved the lower bound from n^{2/3-o(1)} to nearly linear.

Advances understanding of monotone paths in edge-ordered graphs.

Abstract

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 Chv\'atal and Koml\'os 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 $n^{1-o(1)}$.