Determining if two sets are related - that is, if they have similar values or if one set contains the other - is an important problem with many applications in data cleaning, data integration, and information retrieval. A particularly popular metric that has been proposed is to measure the relatedness of two sets by treating the elements as vertices of a bipartite graph and calculating the score of the maximum matching pairing between elements. Compared to other metrics which require exact matchings between elements, this metric uses a similarity function to compare elements between the two sets, making it robust to small dissimilarities in elements and more useful for real-world, dirty data. Unfortunately, the metric suffers from expensive computational cost, taking O(n^3) time, where n is the number of elements in sets, for each set-to-set comparison. Thus for applications which try to search for all pairings of related sets in a brute-force manner, the runtime becomes unacceptably large. To address this challenge, we developed SilkMoth, a system capable of rapidly discovering related set pairs in collections of sets. Internally, SilkMoth creates a signature for each set, with the property that any other set which is related must match the signature. SilkMoth then uses these signatures to prune the search space, so only sets which match the signatures are left as candidates. Finally, SilkMoth applies the maximum matching metric on remaining candidates to verify which of these candidates are truly related sets. Thus, a contribution of this paper is the characterization of the space of signatures which enable this property. We show that selecting the optimal signature in this space is NP-complete, and based on insights from the characterization of the space, we propose two novel filters which help to prune the candidates further before verification.