Title: Pruning Edges and Gradients to Learn Hypergraphs from Larger Sets. Abstract: This paper aims for set-to-hypergraph prediction, where the goal is to infer the set of relations for a given set of entities. This is a common abstraction for applications in particle physics, biological systems and combinatorial optimization. We address two common scaling problems encountered in set-to-hypergraph tasks that limit the size of the input set: the exponentially growing number of hyperedges and the run-time complexity, both leading to higher memory requirements. We make three contributions. First, we propose to predict and supervise the \emph{positive} edges only, which changes the asymptotic memory scaling from exponential to linear. Second, we introduce a training method that encourages iterative refinement of the predicted hypergraph, which allows us to skip iterations in the backward pass for improved efficiency and constant memory usage. Third, we combine both contributions in a single set-to-hypergraph model that enables us to address problems with larger input set sizes. We provide ablations for our main technical contributions and show that our model outperforms prior state-of-the-art, especially for larger sets.