SOLVED

There is a function $f:(1/2,\infty)\to \mathbb{R}$ such that $f(c)\to 0$ as $c\to 1/2$ and $f(c)\to 1$ as $c\to \infty$ and every random graph with $n$ vertices and $cn$ edges has (with high probability) a path of length at least $f(c)n$.

#900: [Er82e]