2-step Nilpotent $L_\infty$-algebras and Hypergraphs

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.