A combinatorial characterization of Kim's lemma for pairs of bi-invariant types

TL;DR

This paper characterizes when Kim's lemma fails for pairs of bi-invariant types using combinatorial configurations, linking model-theoretic properties to graph-theoretic structures and set-theoretic assumptions.

Contribution

It provides a combinatorial framework for understanding Kim's lemma failure for bi-invariant types and connects it to complex graph configurations and set-theoretic principles.

Findings

Failure of Kim's lemma characterized by combinatorial configurations

Existence of parameter families indexed by cographs implies Kim's lemma failure

Certain array configurations lead to failure of stationary local character under GCH

Abstract

We give a combinatorial consistency-inconsistency configuration that is equivalent to the failure of the following form of Kim's lemma for a given $k$: $(\star)$ For any set of parameters $A$, formula $\varphi(x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\mathrm{tp}(b/A)$, if $\varphi(x,b)$ $k$-divides along $p$, then it divides along $q$. We then give an equivalent technical variant of $(\star)$ that is non-trivial over arbitrary invariance bases. We also show that the failure of weaker versions of $(\star)$ entails the existence of stronger combinatorial configurations, the strongest of which can be phrased in terms of families of parameters indexed by arbitrary cographs (i.e., $P_4$-free graphs). Finally, we show that if there is an array $(b_{i,j} : i,j < \omega)$ of parameters such that $\{\varphi(x,b_{i,j}) : (i,j) \in C\}$ is consistent whenever $C \subseteq \omega^2$ is a chain (in the product partial order) and $k$-inconsistent whenever $C$ is an antichain, then there is a model $M$, parameter $b$, and $M$-coheirs $p,q \supset \mathrm{tp}(b/M)$ such that $q^{\otimes \omega}$ is an $M$-heir-coheir and $\varphi(x,b)$ $k$-divides along $p$ but does not divide along $q$. In doing so, we also show that this configuration entails the failure of generic stationary local character under the assumption of $\mathsf{GCH}$.