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Friday, May 4, 2018

**Abstract:** Given a countable family $G$ of analytic hypergraphs on a Polish space $X$, one can consider the quotient poset $P(G)$ of Borel sets modulo the ideal generated by Borel sets which are anticliques in at least one hypergraph in $G$. This is a broad family of forcings, many of them proper. The family is closed under such operations as countable support iteration of (in some cases) product. Most traditional fusion arguments disappear in this class of forcing. They are replaced by simple combinatorial considerations about the edges of the generating hypergraphs.