Some Remarks on Kim-dividing in NATP Theories

This paper establishes that in NATP theories, Kim-dividing over models is always witnessed by coheir Morley sequences, leading to key corollaries regarding quasi-dividing and the characterization of Kim-independence, while also clarifying the relationship between these concepts and Mutchnik's recent work on $\omega$-NDCTP$_2$ theories.

The Gist

The Big Picture: Mapping the Universe of Logic

Imagine the world of mathematical logic (specifically Model Theory) as a vast, foggy landscape. Mathematicians are cartographers trying to draw a map of this landscape to understand how different "theories" (rulesets for how objects interact) behave.

For a long time, they knew about two safe, well-behaved regions:

1. Simple Theories: The "Flatlands." Everything is predictable and orderly.
2. NTP2 and NSOP1: Two distinct "hills" that are slightly more complex but still manageable.

Recently, mathematicians discovered a massive, mysterious region called NATP. Think of this as a giant, dense forest that contains both the NTP2 hill and the NSOP1 hill. It's a huge area where we don't fully understand the rules yet.

The authors of this paper, Joonhee Kim and Hyoyoon Lee, are explorers trying to figure out how to navigate this forest. Specifically, they are looking at a tool called Kim-dividing.

The Core Concept: Dividing vs. Kim-Dividing

To understand what they found, we need to understand the tools they are using: Dividing and Kim-Dividing.

Imagine you have a rule (a formula) that describes a relationship between objects.

• Dividing: This is like checking if a rule breaks when you have a random line of people. If you pick a random sequence of people, does the rule fail? In the "Flatlands" (Simple Theories), if a rule breaks for any random line, it's a problem.
• Kim-Dividing: This is a stricter, more sophisticated test. Instead of a random line, you check if the rule breaks for a special, highly organized line (called an invariant Morley sequence). Think of this as checking if the rule breaks for a line of people who are all wearing identical uniforms and standing in perfect formation.

The Problem: In the big NATP forest, we know that "Dividing" and "Kim-Dividing" are not always the same thing. Sometimes a rule breaks for a random line but works for the uniformed line. This makes navigation hard.

The Main Discovery: The "Coheir" Shortcut

The authors' biggest breakthrough is finding a specific type of "uniformed line" that works perfectly in the NATP forest.

They prove that in the NATP forest, if a rule breaks for a Kim-dividing test (the uniformed line), it must also break for a specific type of line called a Coheir Morley sequence.

The Analogy:
Imagine you are trying to find a leak in a massive dam (the theory).

• Dividing is like throwing a bucket of water at the dam anywhere. If it leaks, you know there's a hole.
• Kim-Dividing is like using a high-pressure hose aimed at a specific, reinforced spot.
• Coheir Morley Sequence is a special kind of hose that is guaranteed to hit the hole if the high-pressure hose would have hit it.

The authors say: "In this specific forest (NATP), you don't need to check every possible uniformed line. If you find a leak using the special 'Coheir' hose, you know for sure the rule is broken."

This is huge because Coheir sequences are easier to find and work with than the other types. It simplifies the map.

The "Tree" Metaphor: Antichain Trees

To prove this, the authors use a complex construction involving trees.

• Imagine a tree where branches split off.
• Some paths go straight down (paths).
• Some branches are side-by-side and never touch (antichains).

In logic, a "bad" theory is one where you can build a tree that has a consistent path down the middle but is chaotic everywhere else. The authors show that in the NATP forest, you cannot build these "bad" trees if you use the Coheir hose. If you try to build a tree that breaks the rules, the forest itself (the NATP property) prevents it from growing.

The "N-ω-DCTP2" Sub-Region

Towards the end, the paper discusses a smaller, even more interesting clearing inside the NATP forest called N-ω-DCTP2.

• This clearing is special because it contains both the NTP2 and NSOP1 hills.
• In this specific clearing, the authors (building on work by Mutchnik) show that Kim-Dividing and Kim-Forking are actually the same thing.

The Analogy:
In the general forest, "Kim-Dividing" (breaking a rule with a uniformed line) and "Kim-Forking" (breaking a rule by combining several broken rules) are different.
But in this specific clearing (N-ω-DCTP2), they are identical. It's like finding a zone in the forest where the laws of physics are simplified, and two different types of magic spells turn out to be the exact same spell.

Why Does This Matter?

1. Simplification: It gives mathematicians a simpler tool (the Coheir sequence) to test theories in this vast NATP region.
2. Unification: It shows that the properties of the two smaller hills (NTP2 and NSOP1) are preserved and extended into this larger forest.
3. New Boundaries: It helps define exactly where the "safe" logic ends and the "chaotic" logic begins, drawing a clearer line on the map.

Summary in One Sentence

The authors discovered that in a vast, complex region of mathematical logic called NATP, a specific, easier-to-use testing method (Coheir sequences) is just as powerful as the more complex method (Kim-dividing), allowing them to map out the rules of this region more clearly and find a special sub-zone where the rules become even simpler.

Technical Summary

1. Problem Statement and Background

Context:
The paper investigates the behavior of Kim-forking and Kim-dividing within the class of NATP theories (theories that do not have the Antichain Tree Property). NATP is a broad class of first-order theories that serves as a common generalization of NTP2 theories (which include simple theories and NTP2) and NSOP1 theories.

The Core Problem:
In well-behaved theories like Simple, NTP2, and NSOP1, there is a strong connection between forking and dividing, often mediated by Kim's Lemma. Kim's Lemma generally states that if a formula divides (or Kim-divides), then it is witnessed by the inconsistency of the formula along every Morley sequence of a specific type (e.g., invariant, strictly invariant, or coheir).

However, in NATP theories, the situation is more complex:

1. Forking is not equivalent to dividing, even over models.
2. It is unclear which class of Morley sequences (invariant, coheir, strictly invariant, etc.) serves as the correct "witness" for Kim-dividing in the general NATP context.
3. Specifically, the authors note that in some NTP2 theories, even coheir Morley sequences may fail to witness Kim-dividing.

Objective:
The authors aim to:

• Determine the relationship between Kim-dividing and coheir-dividing in NATP theories.
• Identify the necessary and sufficient conditions for a Morley sequence to witness Kim-dividing.
• Explore new dividing lines within NATP (specifically related to Mutchnik's recent work on $k$-DCTP2) where stronger equivalences hold.

---

2. Methodology

The authors employ a combination of combinatorial model theory and the analysis of tree properties. Key methodological tools include:

• Tree Properties and Indiscernibility: The paper utilizes the Antichain Tree Property (ATP) and its negation (NATP). They construct specific tree-indexed sets of parameters to derive contradictions if certain independence assumptions fail.
• Type Products and Global Types: They extensively use the tensor product of global types ($p \otimes q$) and the construction of global coheirs and invariant types.
• Induction on Ordinals: A central technique involves constructing sequences of global coheirs indexed by ordinals (specifically using the structure $T^2_\alpha$) to build "antichain trees." This allows them to transfer properties from local sequences to global structures.
• Pre-independence Relations: The authors analyze abstract pre-independence relations (ternary relations satisfying properties like monotonicity, transitivity, and existence) to derive sufficient conditions for the existence of witnesses.
• Comparison with Mutchnik's Work: They integrate and extend recent results by Scott Mutchnik regarding canonical coheirs and the $k$-DCTP2 property.

---

3. Key Contributions and Results

A. Kim-Dividing Implies Coheir-Dividing in NATP

The primary result of the paper is Theorem 3.10:

Theorem: If $T$ is an NATP theory and $M$ is a model, and a formula $\phi(x, a)$ Kim-divides over $M$, then it coheir-divides over $M$.

• Significance: This establishes that in NATP theories, the "stronger" notion of Kim-dividing (witnessed by invariant Morley sequences) implies the "weaker" notion of coheir-dividing (witnessed by coheir Morley sequences).
• Corollaries:
  • Corollary 3.13: If a formula Kim-forks over a model in an NATP theory, it quasi-divides over that model.
  • Corollary 3.17: For any tuple $b$ and model $M$ in an NATP theory, there exists a global coheir $p$ containing $\text{tp}(b/M)$ such that for all $b' \models p|_{MB}$, $B \mathop{\mathpalette\Ind{}}^K_M b'$ (Kim-independence holds).

B. Necessary and Sufficient Conditions for Witnesses

The paper investigates what properties a class of Morley sequences must have to serve as witnesses for Kim-dividing (denoted as condition $Z$ in the paper's uniform statement $[*]$).

• Necessary Condition (Proposition 4.7):
  If a witness of Kim-dividing exists for a tuple $a$ over a model $M$, and a coheir Morley sequence $I$ is a witness, then $I$ must be a strict coheir Morley sequence.

  • Definition: A sequence is a strict coheir if it satisfies a strong form of independence where the sequence is independent from the base in both directions relative to Kim-forking.

• Sufficient Condition (Proposition 4.8):
  The existence of a witness for Kim-dividing is guaranteed if there exists a pre-independence relation $\mathop{\mathpalette\Ind{}}$ that is stronger than heir-independence ($\mathop{\mathpalette\Ind{}}_h$) and satisfies:

  1. Monotonicity.
  2. Full existence.
  3. Right chain condition for coheir Morley sequences.
  4. Strong right transitivity over models.
     If such a relation exists, then Kim-forking is equivalent to Kim-dividing over models.

C. Analysis of N-$\omega$-DCTP2 Theories

The authors analyze a subclass of NATP theories introduced by Mutchnik, called N-$\omega$-DCTP2 (theories without the $k$-descending comb tree property for any $k$).

• Hierarchy: The class N-$\omega$-DCTP2 contains both NTP2 and NSOP1 theories but is strictly contained within NATP.
• Equivalence of Notions:
  • In N-$\omega$-DCTP2 theories, Kim-forking is equivalent to Kim-dividing over models (Remark 5.10).
  • This is derived by combining Theorem 3.10 (Kim-dividing $\implies$ coheir-dividing) with Mutchnik's Fact 5.9 (in N-$\omega$-DCTP2, coheir-dividing $\implies$ canonical coheir-dividing, which implies Kim-dividing).
• Counterexamples: The authors note that in general NATP (and even in N-$\omega$-DCTP2), coheir-Conant-independence is not equivalent to Kim-independence, highlighting the subtlety of these distinctions.

---

4. Significance and Implications

1. Refining the Landscape of NATP: The paper clarifies that while NATP is a broad class, it does not uniformly support the equivalence of forking and dividing. However, it identifies specific subclasses (like N-$\omega$-DCTP2) where these equivalences are restored.
2. Role of Coheirs: The result that Kim-dividing implies coheir-dividing in NATP is a significant step toward understanding the "witness" problem. It suggests that coheir Morley sequences are the natural candidates for witnessing Kim-dividing in this context, provided they satisfy the "strict" condition.
3. Connection to Mutchnik's Work: The paper successfully integrates Mutchnik's recent results on canonical coheirs and DCTP2, showing how these new tree properties create a "sweet spot" within NATP where the independence theory behaves similarly to NSOP1 and NTP2.
4. Future Directions: The paper poses open questions, such as whether there are natural examples of N-$\omega$-DCTP2 theories that possess both TP1 and TP2, and whether NATP is equivalent to N-$\omega$-DCTP2.

Conclusion

Kim and Lee provide a rigorous framework for analyzing Kim-dividing in NATP theories. Their main theorem establishes that Kim-dividing is witnessed by coheir-dividing in this class. Furthermore, they delineate the precise structural requirements (strict coheir sequences, strong right transitivity) needed for the full equivalence of Kim-forking and Kim-dividing, identifying N-$\omega$-DCTP2 as a promising subclass where these strong properties hold.