Cycles as edge intersection hypergraphs of $k$-uniform hypergraphs ($k\le 6$) -- a constructive approach

If $\mathcal{H}=(V,\mathcal{E})$ is a hypergraph, its edge intersection hypergraph $EI(\mathcal{H})=(V,{\mathcal{E}}^{EI})$ has the edge set ${\mathcal{E}}^{EI}=\{e_1\cap e_2\mathrm{\mid }{e}_1,e_2\in \mathcal{E}\wedge e_1\mathrm{\ne }{e}_2\wedge \mathrm{\mid }{e}_1\cap e_2\mathrm{\mid }\ge 2\}$. In the present paper, we consider 4- and 5-uniform hypergraphs $\mathcal{H}$, respectively, with $EI(\mathcal{H})=C_n$. Our results fill the gap between the 3- and the 6-uniform case considered in arXiv:1902.00396.