Edge intersection hypergraphs - a new hypergraph concept

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph, its edge intersection hypergraph $EI(\mathcal{H})=(V,{\mathcal{E}}^{EI})$ has the edge set ${\cal E}^{EI}={e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq2}$. Besides investigating several structural properties of edge intersection hypergraphs, we prove that all trees but seven exceptional ones are edge intersection hypergraphs of 3-uniform hypergraphs.