Existence conjecture for independent-set densities in sparse random graphs

Let ${\bf {\cal I}}(n,c)$ denote the maximum size of an independent set in the Erdős–Rényi random graph $G(n,c/n)$, and let ${\bf {\cal I}}(n,r)$ denote the maximum size of an independent set in a random $r$-regular graph. The parameters satisfy $c>0$ and $r\geq 3$.

Independent-set density existence conjecture. For every $c>0$ and $r\geq 3$, the limits

$$\lim_{n\rightarrow\infty}{\mathbb{E}[{\bf {\cal I}}(n,c)]\over n}, \qquad \lim_{n\rightarrow\infty}{\mathbb{E}[{\bf {\cal I}}(n,r)]\over n}$$

exist.

The conjecture concerns the existence of limiting expected independent-set densities in both sparse Erdős–Rényi and random regular graphs. The supplied text says that it remains a conjecture and attributes it to prior work by Aldous and Steele and to Aldous's collection of open problems.