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 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 $n^{2\mathrm{/}3-o(1)}$. In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length $n^{1-o(1)}$.