Kim’s Lemma is a key ingredient in the theory of forking independence in simple
theories. It asserts that if a formula divides, then it divides along every Morley sequence
in type of the parameters. Variants of Kim’s Lemma have formed the core of the theories
of independence in two orthogonal generalizations of simplicity — namely, the classes
of
and
theories. We introduce a new variant of Kim’s Lemma that simultaneously generalizes
the
and
variants. We explore examples and nonexamples in which this lemma holds,
discuss implications with syntactic properties of theories, and ask several
questions.
