Petrykowski’s bounded-type-orbit implication to definable amenability

If a definable group has a global type with bounded orbit under left translation, must it be definably amenable, meaning that it has a left-invariant Keisler measure? The question allows theories outside NIP.

Why it matters and what counts as progress

Why it matters: It tests whether bounded type dynamics can imply invariant measure existence without the tameness assumptions in known positive results.

What counts as progress: A general implication or a definable group with a bounded-orbit type and no invariant Keisler measure, retaining the theory and boundedness assumptions.

How the problem developed

2012 · Conjecture [1]
Ludomir Newelski Published the bounded-orbit framework and the conjecture, attributed to Petrykowski, that a definable group with a bounded type orbit is definably amenable.
2018 · Partial result [2]
Artem Chernikov and Pierre Simon Proved the bounded-orbit implication for groups definable in NIP theories.
4 Sept 2026 · Counterexample [3]
Artem Chernikov Constructed a complete simple theory with a global type of orbit size at most 2ℵ₀ in a group with no left-invariant Keisler measure, refuting the unrestricted implication.

AI contribution

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.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFull