  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 $f(K_n) \ge \sqrt{n - 3/4} - 1/2$ and in 1984,
Calderbank, Chung, and Sturtevant proved that $f(K_n) \le (\frac{1}{2} +
o(1))n$. We show that $f(K_n) \ge (\frac{1}{20} - o(1))(n/\lg n)^{2/3}$.
