The connectivity threshold conjecture for uniform random intersection graphs

Let $G(n,m,k)$ be the random intersection graph on $n$ vertices in which each vertex is assigned a uniformly random $k$-subset of a colour set of size $m$, with two vertices adjacent when their assigned colour sets intersect. Let $k$ and $m$ be functions of $n$, and let $\omega\to\infty$ as $n\to\infty$.

Connectivity threshold conjecture.

(i) If

$$\frac{k^2n}{m}=\log n+\omega,$$

then almost surely $G(n,m,k)$ is connected. (ii) If

$$\frac{k^2n}{m}=\log n-\omega,$$

then almost surely $G(n,m,k)$ is not connected.

This conjecture predicts a sharp connectivity threshold for uniform random intersection graphs at $k^2n/m=\log n$.