Chang's Conjectures are strengthenings of the Lowenheim-Skolem
theorem. Given an arbitrary structure A they ask for an elementary
substructure B where the cardinalities of the intersections of B with
various predicates are specified in advance.
If A is well-founded one could also ask that the transitive collapse of B
be large; i.e. that B have strong condensation properties. In this talk a
strong Chang's Conjecture of this form is presented. The consistency
strength of this Chang's Conjecture holding at omega_3 is between a
2-huge and a huge cardinal.