Monotone Paths in Dense Edge-Ordered Graphs

The altitude of a graph $G$, denoted $f(G)$, is the largest integer $k$ such that under each ordering of $E(G)$, there exists a path of length $k$ which traverses edges in increasing order. In 1971, Chv'atal and Koml'os asked for $f(K_n)$, where $K_n$ is the complete graph on $n$ vertices. In 1973, Graham and Kleitman proved that $f(K_n)\ge \sqrt{n-3\mathrm{/}4}-1\mathrm{/}2$ and in 1984, Calderbank, Chung, and Sturtevant proved that $f(K_n) \le (\frac{1}{2} + o(1))n$\mathrm{.}Weshowthat$f(K_n) \ge (\frac{1}{20} - o(1))(n/\lg n)^{2/3}$.