Some applications of the real strict order property hierarchy

Abstract: We give applications of the properties $\mathrm{NSOP}<em>{r}$ for non-integer values of $r$ to problems on the original hierarchy $\mathrm{NSOP}</em>{n}$ for integer values of $n$. We first show that the properties $\mathrm{NSOP}<em>{r}$, previously defined for real values $r \geq 3$, are even well-defined for real values $r \geq 2$, showing that $$\mathrm{NSOP}</em>{2} \subseteq \mathrm{NSOP}<em>{r}$$ for our original definition of $\mathrm{NSOP}</em>{r}$ even when $2 < r < 3$. As a consequence, newness of all of the well-defined properties $\mathrm{NSOP}<em>{r}$ for non-integer $r$ would negatively resolve the problem of whether $\mathrm{NSOP}</em>{2}$ is equal to $\mathrm{NSOP}<em>{3}$. We then prove an approximate alternative between two possibilities: (1) that in extending Shelah's original $\mathrm{NSOP}</em>{n}$ hierarchy for integers $n \geq 3$ to the $\mathrm{NSOP}<em>{r}$ hierarchy for reals $r &gt; 2$, we really did introduce new classification-theoretic properties, and (2) that $$\mathrm{NSOP}</em>{n+1} \cap \mathrm{NTP}<em>{2} = \mathrm{NSOP}</em>{n} \cap \mathrm{NTP}<em>{2}$$ for integers $n \geq 3$, which would resolve a central open problem in classification theory. More precisely, we give a rigorous sense in which (1) can fail on particularly general grounds, and then show that if (1) fails for these general reasons, (2) must be true. Finally, we apply cycle-removal techniques from the theory of the properties $\mathrm{NSOP}</em>{r}$ for real-values of $r$ to make progress on the question of whether $\mathrm{NSOP}<em>{2}$ is equal to $\mathrm{NSOP}</em>{3}$. We (a) show that if $\mathcal{H}$ is a hereditary class of structures defined by finitely many forbidden weakly embedded substructures, if every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}<em>{2}$, then every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}</em>{3}$, and (b) observe that we cannot replace $\mathrm{SOP}_{2}$ with $\mathrm{TP}$ here.

• The paper introduces a real-valued NSOP_r hierarchy (r > 2) that refines traditional NSOP_n classifications using advanced combinatorial and independence-theoretic techniques.
• It establishes a dichotomy showing that either non-integer NSOP_r properties are genuinely new or the integer-valued NSOP_n hierarchy collapses within NTP_2 theories.
• The findings offer practical insights into the structure of models, with applications in graph theory, automorphism groups, and constraint satisfaction problems.

Summary of "Some applications of the real strict order property hierarchy" (2606.28740)

Context and Motivation

The paper extends the analysis of the strict order property (SOP) hierarchy in model theory, particularly focusing on properties $\mathrm{NSOP}_n$ (negations of the $n$-strict order property) and, crucially, on developing and analyzing a real-valued generalization $\mathrm{NSOP}_r$ for real values $r>2$. The distinction between $\mathrm{NSOP}_2$ and $\mathrm{NSOP}_3$ and, more generally, the strictness of the $\mathrm{NSOP}_n$ hierarchy within $\mathrm{NTP}_2$ theories, are longstanding classification-theoretic open problems. The paper's core aim is to rigorously justify and exploit the real-valued $\mathrm{NSOP}_r$ hierarchy and demonstrate how it yields both theoretical insight and combinatorial applications to longstanding open problems involving stability, order, and the tree property.

Main Contributions

Extension and Formalization of the $\mathrm{NSOP}_r$ Hierarchy

The author defines $n$0 for any real number $n$1, generalizing Shelah’s original integer-indexed $n$2 hierarchy. This extension is justified via a careful combinatorial and independence-theoretic argument that shows the hierarchy remains a (potentially proper) ascending chain: $n$3 for $n$4. The proof uses the equality $n$5 (proven in previous work [NSOP2]) and leverages Kim-independence tools in $n$6 theories [KR17].

Approximate Alternatives and Hierarchy Strictness

A core result is the formulation of an approximate alternative: either the real-valued $n$7 introduces genuinely new properties distinct from the integer-valued $n$8 (for $n$9 non-integer), or the intersection hierarchy $\mathrm{NSOP}_r$0 collapses at each level for $\mathrm{NSOP}_r$1. The paper constructs a combinatorial framework where this dichotomy becomes apparent and shows that, under a "nondistinctness on sufficiently general grounds" hypothesis, the equality $\mathrm{NSOP}_r$2 follows by simulating standard arguments about generic cycle-free graphs and the $\mathrm{NSOP}_r$3-tree property.

Combinatorial Progress on $\mathrm{NSOP}_r$4 vs $\mathrm{NSOP}_r$5

The most substantial technical development is a sharp dichotomy for hereditary classes defined by finitely many forbidden weakly embedded substructures: if every theory whose models have age $\mathrm{NSOP}_r$6 has $\mathrm{NSOP}_r$7, then all such theories also have $\mathrm{NSOP}_r$8. This result is achieved via a highly technical cycle-removal strategy on covering maps (helix maps) and is robust in that the corresponding dichotomy replacing $\mathrm{NSOP}_r$9 by the tree property fails as shown by concrete counter-examples. The proof integrates combinatorial results on hereditary classes, model companions, and structural analysis of witnesses to $r>2$0 with careful handling of algebraic closures and Morley sequences.

Further Applications

The techniques developed enable applications to:

• Graphs definable in $r>2$1 theories: any such graph omitting some weakly embedded subgraph must, for all $r>2$2, omit a weakly embedded graph with no $r>2$3-cycles.
• Approximate implications between classification-theoretic properties, formalizing notions like $r>2$4 in terms of obstructions at the finite level.

Strong/Contradictory Claims

• The extension of $r>2$5 to all reals $r>2$6 forms a consistent and well-defined hierarchy, settling technical doubts regarding the preservation of the ascending chain beyond integer parameters.
• The paper’s main dichotomy, under minimal hereditary class hypotheses, asserts that $r>2$7 implies $r>2$8 for all such $r>2$9—contradicting any naive intuition that the gap between $\mathrm{NSOP}_2$0 and $\mathrm{NSOP}_2$1 could be realized within these settings.
• The formulation of an "approximate alternative" on the strictness of the classification-theoretic hierarchy provides a blueprint for resolving the strictness problem via combinatorial (as opposed to purely model-theoretic) techniques.

Theoretical and Practical Implications

Theoretical Framework

These results significantly clarify the landscape of unstable classification theory and its combinatorial underpinnings. The real-valued $\mathrm{NSOP}_2$2 hierarchy offers a parametrized stratification, expanding the toolkit for distinguishing and analyzing dividing lines between stability, simplicity, the SOP hierarchy, and the tree property. The combinatorial tools for hereditary classes connect infinite combinatorics, graph theory, and model theory, and provide templates for future negative and positive answers to hierarchy strictness.

Applications and Future Developments

• Classification Theory: The new hierarchy serves as a lens for re-examining existing open problems about the strictness of dividing lines and the equationality of various classes (e.g., the ongoing question of whether $\mathrm{NSOP}_2$3).
• Combinatorial Model Theory: The cycle-removal and helix map arguments developed here will likely be transferable to related questions about finite and infinite combinatorial configurations implicated in model-theoretic properties.
• Automorphism Groups and CSPs: The connection with model companions and the preservation of $\mathrm{NSOP}_2$4 points toward applications in the analysis of automorphism groups, generic structures, and constraint satisfaction, especially in the context of structures defined by forbidden patterns.
• Potential for Finer incompactness phenomena: The link between real and integer indices could suggest analogous incompactness properties in continuous logic or for other real-parameterized combinatorial invariants.

Conclusion

This paper advances the theory of classification by rigorously defining and applying a real-valued extension of the strict order property hierarchy, connecting combinatorial, algebraic, and independence-theoretic perspectives. It provides new unconditional structural results on theory classes defined by forbidden configurations, resolves technical obstacles in hierarchy extensions, and aligns longstanding open questions within a broader combinatorial framework. The techniques and results should be foundational for further developments in the theory of dividing lines, unstable hierarchies, and the combinatorics of definability in model theory.