Given a hypergraph $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$.

A *linear hypergraph* is a hypergraph $H=(V,E)$ such that every edge has at least $2$ elements and for all $e\neq e'\in E$ we have $|e\cap e'|\leq 1$.

It turns out that for every simple undirected graph $G$, finite or infinite, there is a linear hypergraph $H$ such that $I(H)\cong G$. The *linear intersection number* $\ell(G)$ of a graph $G$ is the smallest cardinal $\kappa$ such that there is a linear hypergraph $H=(\kappa,E)$ such that $I(H)\cong G$.

**Question.** Is there an infinite graph $G$ such that $\ell(G) < \chi(G)$?

**Note.** For finite graphs, the answer to this question is not known.