Equivalence of SOP₂ and SOP₃
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SOP₂ and SOP₃ are combinatorial dividing lines for complete first-order theories. SOP₃ is known to imply SOP₂; does SOP₂ also imply SOP₃?
Why it matters and what counts as progress
Why it matters: Equivalence would collapse two proposed levels in the classification of unstable theories.
What counts as progress: A proof of the reverse implication for every complete theory, or a complete theory satisfying SOP₂ while failing SOP₃.
How the problem developed
- 2004 · Open problem [1]
Márta Džamonja and Saharon Shelah Asked whether the established implication from SOP₃ to SOP₂ can be reversed.
- 13 Aug 2026 · Theorem [1]
Artem Chernikov Proved that SOP₂ and SOP₃ define the same class of first-order theories, establishing the reverse implication and resolving the question.
AI contribution
SOP₂ = SOP₃
For every complete first-order theory, SOP₂ implies SOP₃. The reverse implication was already known, so the two dividing lines coincide.
Sources
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