Large cardinal axioms from tameness in aecs

We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions. For instance, we show that if κ is an uncountable cardinal such that μ^ω <κ for every μ<κ and every AEC with Löwenheim-Skolem number less than κ is <κ-tame, then κ is almost strongly compact. This is done by isolating a class of AECs that exhibits tameness exactly when sufficiently complete ultrafilters exist.