### Categoricity and multidimensional diagrams

by Shelah and Vasey. [ShVe:842]

We study multidimensional diagrams in independent amalgamation in
the framework of abstract elementary classes (AECs). We use them to
prove the eventual categoricity conjecture for AECs, assuming a
large cardinal axiom. More precisely, we show assuming the existence
of a proper class of strongly compact cardinals that an AEC which has
a single model of emph {some} high-enough cardinality will have a
single model in emph {any} high-enough cardinal.
Assuming a weak version of the generalized continuum hypothesis, we
also establish the eventual categoricity conjecture for AECs with
amalgamation.

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