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Petrykowski's Conjecture

Petrykowski's conjecture (Newelski 2012) asserted that a definable group G is definably amenable whenever it admits a global type p in S_G(U) with bounded orbit under left translation. Here bounded means that the orbit's cardinal number is smaller than the saturation cardinal of the monster model U.

The conjecture holds for groups in theories without the independence property (Chernikov and Simon 2018), but fails in general. Chernikov (2026) constructed a complete simple theory, a definable group G, and a global type p such that |G·p|<=2^(aleph_0), while G has no left-invariant Keisler measure. The counterexample is a simple expansion of the theory of nonabelian free groups.

Chernikov (2026) reports that ChatGPT 5.6 established parts of the general construction and found the bounded-orbit type construction. The author subsequently simplified and reorganized the arguments. Independent external review had not been reported as of Sep. 23, 2026.

See also

Definable Amenability, Definable Group, Global Type, Independence Property, Keisler Measure, Model Theory, Simple Theory

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References

Chernikov, A. "A Counterexample to Petrykowski's Conjecture." 4 Sep 2026. https://arxiv.org/abs/2609.05711.Chernikov, A. and Simon, P. "Definably Amenable NIP Groups." J. Amer. Math. Soc. 31, 609-641, 2018. https://doi.org/10.1090/jams/896.Newelski, L. "Bounded Orbits and Measures on a Group." Israel J. Math. 187, 209-229, 2012. https://doi.org/10.1007/s11856-011-0081-x.

Cite this as:

Weisstein, Eric W. "Petrykowski's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PetrykowskisConjecture.html

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