A combinatorial characterization of Kim's lemma for pairs of bi-invariant types
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916868211605504 |
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| author | Hanson, James E. |
| author_facet | Hanson, James E. |
| contents | 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 $φ(x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\mathrm{tp}(b/A)$, if $φ(x,b)$ $k$-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 $(b_{i,j} : i,j < ω)$ of parameters such that $\{φ(x,b_{i,j}) : (i,j) \in C\}$ is consistent whenever $C \subseteq ω^2$ 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 $q^{\otimes ω}$ is an $M$-heir-coheir and $φ(x,b)$ $k$-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 $\mathsf{GCH}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A combinatorial characterization of Kim's lemma for pairs of bi-invariant types Hanson, James E. Logic 03C45 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 $φ(x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\mathrm{tp}(b/A)$, if $φ(x,b)$ $k$-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 $(b_{i,j} : i,j < ω)$ of parameters such that $\{φ(x,b_{i,j}) : (i,j) \in C\}$ is consistent whenever $C \subseteq ω^2$ 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 $q^{\otimes ω}$ is an $M$-heir-coheir and $φ(x,b)$ $k$-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 $\mathsf{GCH}$. |
| title | A combinatorial characterization of Kim's lemma for pairs of bi-invariant types |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2507.21366 |