Real-valued NSOP_r techniques yield that NSOP2subseteq NSOP_r for r>2, an approximate alternative between new real properties and NSOP_n collapse in NTP2, and a sharp SOP2-implies-SOP3 dichotomy for finitely forbidden weak-embedding hereditary classes.
Properties of independence in $\mathrm{NSOP}_3$ theories
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abstract
We prove some results about the theory of independence in $\mathrm{NSOP}_{3}$ theories that do not hold in $\mathrm{NSOP}_{4}$ theories. We generalize Chernikov's work on simple and co-simple types in $\mathrm{NTP}_{2}$ theories to types with $\mathrm{NSOP}_{1}$ induced structure in $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ and $\mathrm{NSOP}_{3}$ theories, and give an interpretation of our arguments and those of Chernikov in terms of the characteristic sequences introduced by Malliaris. We then prove an extension of the independence theorem to types in $\mathrm{NSOP}_{3}$ theories whose internal structure is $\mathrm{NSOP}_{1}$. Additionally, we show that in $\mathrm{NSOP}_{3}$ theories with symmetric Conant-independence, finitely satisfiable types satisfy an independence theorem similar to one conjectured by Simon for invariant types in $\mathrm{NTP}_{2}$ theories, and give generalizations of this result to invariant and Kim-nonforking types.
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math.LO 2years
2026 2representative citing papers
Authors prove SOP_n is straightly definable and poset definable for n >= 4, completing the straight definability classification of classical model-theoretic properties, and show that implications between positively straightly definable properties in countably categorical theories occur at the exists
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Some applications of the real strict order property hierarchy
Real-valued NSOP_r techniques yield that NSOP2subseteq NSOP_r for r>2, an approximate alternative between new real properties and NSOP_n collapse in NTP2, and a sharp SOP2-implies-SOP3 dichotomy for finitely forbidden weak-embedding hereditary classes.
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On the notion of a patterning property in model theory
Authors prove SOP_n is straightly definable and poset definable for n >= 4, completing the straight definability classification of classical model-theoretic properties, and show that implications between positively straightly definable properties in countably categorical theories occur at the exists