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A counterexample to Petrykowski's conjecture

Artem Chernikov

Chernikov constructs a complete simple theory with a definable group and a global type whose left-translation orbit has size at most 2^ℵ₀, although the group has no left-invariant Keisler measure. This refutes the bounded-orbit implication to definable amenability outside NIP.

Sources for this record

What was already known

Newelski proved the implication under stronger absolute boundedness, and later work established it for NIP and o-minimal groups. An earlier non-amenable simple-theory example did not supply the bounded-orbit type.

AI involvement AI-centered

Chernikov credits ChatGPT 5.6 with parts of the model-companion construction and discovery of the bounded-orbit type. He simplified the argument, connected it to earlier joint work and wrote the final paper.

The model supplied central construction steps inside Chernikov's human-led extension of earlier work.

Importance Notable

The example settles the proposed implication negatively, even in a simple theory.

Limitations

The paper does not document a completed check of its full construction. The AI disclosure assigns parts of the model-companion construction and the bounded-orbit type to ChatGPT without mapping every lemma.

Technical verification Reported

The v1 paper provides a proof, but its disclosure documents simplification and author responsibility rather than an explicit completed proof check or machine formalization. Reported is the evidence default, not evidence of a failed check.

Human understanding Full

Chernikov says he simplified and streamlined the model-aided arguments, and the paper explains how the model companion, good coloring and finite-quotient construction combine to create the bounded orbit.

Correction and publication history