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There exists a rich set of relations between entanglement entropies of a multi-partite quantum system, some applying universally to all quantum systems (such as strong sub-additivity), some pertaining to some limited but interesting subclass of systems (such as monogamy of mutual information in holography). We focus on QFTs with a holographic dual describing a classical geometry in the bulk, and discuss ongoing work aiming to obtain new entanglement relations for such systems, which should ultimately provide a construction of the multi-partite holographic entropy cone.