Simple Homogeneous Structures and Indiscernible Sequence Invariants

We introduce some properties describing dependence in indiscernible sequences: Find and its dual FMb, 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 Find (equal in DCF₀), 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 FMb(q) = ∞, and on n-resolvability. We prove some variants of the simple Kim-forking conjecture, a generalization of the stable forking conjecture to NSOP₁ theories. We show a global analogue of the simple Kim-forking conjecture with infinitely many variables holds in every NSOP₁ theory, and show that Kim-forking with a realization of a type p with FMb(p) < ∞ satisfies a finite-variable version of this result. We then show, in a low NSOP₁ 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 FMb(p) < ∞ and |S^FMb(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 FMb, motivated in simple and NSOP₁ theories by the above results, is in fact nontrivial even in stable theories.