The Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture can be stated in a very similar form:

**Erdös-Faber-Lovasz conjecture**: for all finite simple undirected graphs $G=(V,E)$ we have $\chi(G)\leq \ell(G)$ where $\ell(G)$ is the linear intersection number (which we define below). (This article shows why this is equivalent to the original statement.)

**Hadwiger's conjecture**: for all finite simple undirected graphs $G=(V,E)$ we have $\chi(G)\leq \eta(G)$ where $\eta(G)$ is the largest $n\in \omega$ such that $K_n$ is a minor of $G$.

**Question**: We would have an implication between the two conjectures if either $\ell(G) \leq \eta(G)$ for all graphs $G$ or $\ell(G) \geq \eta(G)$ for all graphs $G$. Does any of these inequalities hold for all graphs?

A *linear hypergraph* is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties:

- for $e\in L$ we have $|e|\geq 2$;
- if $e_1\neq e_2 \in L$ then $|e_1\cap e_2|\leq 1$.

We set $X(\pi)=X$ and $L(\pi)=L$. The *graph* $G_\pi$ *associated to* a linear hypergraph $\pi$ is given by $G=(V,E)$ where $V = L$ and $E = \{\{e_1, e_2\} \subseteq L: e_1\neq e_2\text{ and } e_1\cap e_2\neq \emptyset\}$. It turns out that for any graph $G$ there is a linear hypergraph $\pi$ such that $G\cong G_\pi$. For any graph $G$ the we set $$\ell(G) := \text{min}\{|X(\pi)|:\pi \text{ is a linear hypergraph such that } G_{\pi} \cong G\}$$ and call this the *linear intersection number* of $G$.