If ${\cal C}$ is a collection of subsets of a set $X$, we associate to ${\cal C}$ a graph $G_{\cal C} = (V,E)$ where $V = {\cal C}$ and $$E = \big\{\{A,B\}: A\neq B\in {\cal C} \land A\cap B \neq \emptyset\big\}.$$

If $G$ is a simple, undirected graph, we define its *intersection number* $\iota(G)$ to be the smallest cardinal $\kappa$ such that there is a collection ${\cal C}$ of subsets of $\kappa$ such that $G_{\cal C} \cong G$.

For any cardinal $\kappa>0$ we define $\log(\kappa) = \min\{\mu\in\kappa\cup\{\kappa\}: 2^\mu \geq \kappa\}.$

If $G=(V,E)$ is infinite, is it true that $\iota(G) \in \{\;\log(|V|), |V| \;\}$?