Let T be a countable complete theory and let P be a distinguished unary predicate in its vocabulary. Assume that T fails the Gaifman property; that is, assume that there exists a model N of the theory of P such that no model M models T satisfies P^M=N. Then for every regular cardinal lambda big enough, and every mu≥lambda, T has 2^lambda models of cardinality mu which are non-isomorphic over P.

Generalized Gaifman Conjecture

The conjecture, associated with Shelah and Usvyatsov, predicts that failure of the Gaifman property forces many non-isomorphic models over a distinguished predicate. Its instability-to-nonstructure direction was previously open.

Known results

Under suitable stability over $P$ , including $n$ -stability for every $n<\omega$ and very stable embeddedness of $P$ , the Gaifman property holds (2025).
The remaining implication was formulated as Conjectures 7.6 and 7.7 and left open (2025).

August 2026 claimed counterexample

Yi Zhang’s preprint claims a negative answer. It constructs a complete stable theory in the finite relational language $L=\{P,E\}$ with, for every infinite cardinal $\kappa$ , a model $N$ satisfying $I_T(\kappa,N)=0$ while $I_T(\kappa,\kappa)=\kappa$ ; a second construction gives further failures of the predicted model-counting behavior. The claim is unverified.

Current status (as of August 2026): The conjecture has a published partial positive theorem and a new, unverified preprint claiming counterexamples; independent validation is pending.

Generalized Gaifman Conjecture

Primary source

Additional references

Progress summary

Solutions 0