Monotone Paths in Dense Edge-Ordered Graphs

TL;DR

This paper investigates the altitude of complete graphs under edge orderings, establishing a new lower bound that grows as a fractional power of n over log n, improving previous bounds.

Contribution

It provides a novel lower bound for the altitude of complete graphs, showing it grows as (n/ log n)^{2/3}, advancing understanding of monotone paths in dense edge-ordered graphs.

Findings

Established a lower bound of (1/20 - o(1))(n/ log n)^{2/3} for f(K_n)

Improved previous bounds on the length of monotone paths in dense graphs

Connected the problem to combinatorial properties of edge orderings in complete graphs.

Abstract

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 $f(K_n) \ge (\frac{1}{20} - o(1))(n/\lg n)^{2/3}$.