Is symmetric/independent forking independence a sufficient condition for simplicity?

Question

I was looking through the paper On Kim-Independence for a masters project and two of the results were that symmetric/independent Kim-independence over models is a sufficient condition for a theory to be NSOP$_1$.

I was wondering if there were similar results for forking and simple theories without the full conditions for the Kim-Pillay theorem - ie. if $T$ is a complete theory, and forking independence is either symmetric or satisfies the independence theorem over models, then $T$ is a simple theory.

Answer 1

Yes, see "Simplicity, and stability in there" by Kim, Journal of Symbolic Logic, 66(2):822-836 (https://doi.org/10.2307/2695047). In particular, Theorem 2.4 in that paper is as follows.

Theorem 2.4. Let $T$ be arbitrary [i.e. any complete first-order theory]. The following are all equivalent.

1. Forking (Dividing) satisfies local character.
2. Forking (Dividing) satisfies symmetry.
3. Forking (Dividing) satisfies transitivity.
4. A formula $\varphi(x, a)$ divides (forks) over a set $A$ if and only if for any Morley sequence $I$ of $\operatorname{tp}(a/A)$, $\{\varphi(x, a') : a' \in I\}$ is inconsistent.

A variant of the fourth condition for NSOP$_1$ is also present in the paper by Kaplan and Ramsey that you cite. That condition is called "Kim's lemma" there and the equivalence with NSOP$_1$ is Theorem 3.16 in that paper. A variant of the third condition is also true for NSOP$_1$ theories, but this can be found in a different paper, see Corollary 4.4 in "Transitivity of Kim-independence" by Kaplan and Ramsey, Advances in Mathematics 379 (https://doi.org/10.1016/j.aim.2021.107573). Finally, a version of the first condition is also in the paper you cite (Corollary 4.6), but I prefer "Local character of Kim-independence" by Kaplan, Ramsey and Shelah, Proceedings of the American Mathematical Society, 147(4):1719-1732 (https://doi.org/10.1090/proc/14305), where it is Theorem 1.1.

So combining these things we get for NSOP$_1$ theories the following version, where you the above references can be used to chase up exact statements.

Theorem. Let $T$ be any complete first-order theory. The following are all equivalent.

1. $T$ is NSOP$_1$.
2. Kim-independence satisfies local character.
3. Kim-independence satisfies symmetry.
4. Kim-independence satisfies transitivity.
5. Kim's lemma for Kim-dividing is holds: for any $M \models T$ and any $\varphi(x, b)$, if $\varphi(x, y)$ $q$-divides for some global $M$-invariant $q \supseteq \operatorname{tp}(b/M)$, then $\varphi(x, y)$ $r$-divides for every global $M$-invariant $r \supseteq \operatorname{tp}(b/M)$.

Note that in all these statements we restrict Kim-independence to models in the base.