A hypergraph G = (V , E) is (k, )-sparse if no subset V ⊂ V spans more than k|V | − hyperedges. We characterize (k, )sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical

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... ior in terms of the sparsity parameters k and . Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425-1437] from graphs to hypergraphs.