# Linear intersection number and chromatic number for infinite graphs

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.

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 $$|E|\leq\sum_{\alpha<\kappa}|E_\alpha|\leq\kappa^2=\kappa.$$