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

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 $\phi (x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\mathrm{tp}(b/A)$, if $\phi (x,b)$ $k$-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 $\{\phi (x,b_{i,j}):(i,j)\in C\}$ 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 $\phi (x,b)$ $k$-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 $\mathsf{GCH}$.

25 pages, 8 figures