Given a family of (hyper)graphs F a (hyper)graph G is said to be F-saturated if G is F-free for any F but for any edge e in the complement of G the (hyper)graph $G + e$ contains some F. We survey the problem of determining the minimum size of an F-saturated (hyper)graph and collect many open problems and conjectures.
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Faudree et al. (2011) studied this question.
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