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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.

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The answer is no. This is equivalent to stating that for any linear hypergraph $H=(\kappa,E)$ with $\kappa$ infinite (note that for $\kappa$ finite, $G=I(H)$ would be finite), we have $\chi(I(H))\leq\kappa$. This follows immediately once we show $|V(I(H))|=|E|\leq\kappa$.

For $\alpha<\kappa$ let $E_\alpha=\{e\in E:\alpha\in e\}$. If we remove $\alpha$ from every element of $E_\alpha$ we get a family of disjoint subsets of $\kappa$. By considering their least elements, it's clear there are at most $\kappa$-many of them, i.e. $|E_\alpha|\leq\kappa$. Therefore $$|E|\leq\sum_{\alpha<\kappa}|E_\alpha|\leq\kappa^2=\kappa.$$

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