2635. John Baldwin, James Freitag and Scott Mutchnik Simple Homogeneous Structures and Indiscernible Sequence Invariants
E-mail:
Submission date: 13 May 2024
Abstract:
We introduce some properties describing dependence in indiscernible
sequences: F_{ind} and its dual F_{Mb}, the definable Morley property, and
n-resolvability. Applying these properties, we establish the following
results:
We show that the degree of nonminimality introduced by Freitag and Moosa,
which is closely related to F_{ind} (equal in DCF_0), may take
on any positive integer value in an ω-stable theory, answering a
question of Freitag, Jaoui, and Moosa.
Proving a conjecture of Koponen, we show that every simple theory with
quantifier elimination in a finite relational language has finite rank and is
one-based. The arguments closely rely on finding types q with F_{Mb}(q) =
∞, and on n-resolvability.
We prove some variants of the simple Kim-forking conjecture, a generalization
of the stable forking conjecture to NSOP_1 theories. We show a
global analogue of the simple Kim-forking conjecture with infinitely many
variables holds in every NSOP_1 theory, and show that Kim-forking
with a realization of a type p with F_{Mb}(p) < ∞ satisfies a
finite-variable version of this result. We then show, in a low
NSOP_1 theory or when p is isolated, if p ∈ S(C) has the
definable Morley property for Kim-independence, Kim-forking with realizations
of p gives a nontrivial instance of the simple Kim-forking conjecture itself.
In particular, when F_{Mb}(p) < ∞ and |S^{F_{Mb}(p) + 1}(C)| < ∞,
Kim-forking with realizations of p gives us a nontrivial instance of the
simple Kim-forking conjecture.
We show that the quantity F_{Mb}, motivated in simple and
NSOP_1 theories by the above results, is in fact nontrivial even
in stable theories.