Counterexamples to the Generalized Gaifman Conjecture

Abstract: Shelah and Usvyatsov proposed the Generalized Gaifman Conjecture [SU25, Conjecture 1.1]. We give a negative answer.

• The paper refutes the Generalized Gaifman Conjecture by constructing theories where failure of the Gaifman property yields far fewer than 2^κ extensions, including 0 or exactly κ extensions over suitable bases.
• The paper’s stable discrete example adds one element to an equivalence-relation structure, while its dense-order example identifies extensions with cuts and achieves I_T(κ,κ)=ded(κ).
• These constructions show that extension counts can vary widely, including consistently intermediate values between κ and 2^κ, while leaving uniform cardinal bounds as an open problem.

The Generalized Gaifman Conjecture, proposed by Shelah and Usvyatsov, predicts that failure of the Gaifman property for a complete theory $T$ with a distinguished unary predicate $P$ forces many pairwise non-isomorphic extensions over a common $P$-part — specifically $2^\kappa$ models of size $\lambda \geq \kappa$, for all sufficiently large regular $\kappa$ (2608.16099). The paper under review refutes this conjecture by exhibiting explicit counterexamples. The constructions are strikingly simple: in both cases, every model of $T$ is obtained from its $P$-part by adding a single element, so the number of extensions over a fixed base is governed directly by combinatorial features of the base structure. The paper also notes that the author found the proof with the assistance of ChatGPT, taking full responsibility for the content.

Background and the conjecture

Fix a countable relational language $L$, a complete first-order $L$-theory $P$0, and a distinguished unary predicate $P$1. The Gaifman property asserts that every model of $P$2 occurs as the $P$3-part of some model of $P$4; relative categoricity over $P$5 asserts that two models with the same $P$6-part are isomorphic over it. Gaifman's original 1974 question — whether relative categoricity implies the Gaifman property — remains open, as Pillay's recent survey attests.

Shelah and Usvyatsov strengthened this to a quantitative claim: for every sufficiently large regular $P$7 and every $P$8, failure of the Gaifman property should yield $P$9 models of cardinality $P$0 that are pairwise non-isomorphic over a common $P$1-part. The paper works with the invariant $P$2, the number of models of $P$3 of size $P$4 with $P$5-part $P$6, taken up to isomorphism fixing $P$7 pointwise, and $P$8 as the supremum over bases $P$9 of size $2^\kappa$0 (replacing Shelah–Usvyatsov's displayed maximum with a supremum).

The discrete counterexample

The first construction is a complete stable theory $2^\kappa$1 in the finite language $2^\kappa$2, where $2^\kappa$3 is an equivalence relation on a countable set $2^\kappa$4 with infinitely many classes of each finite size $2^\kappa$5 and one infinite class $2^\kappa$6; the predicate $2^\kappa$7 is interpreted as $2^\kappa$8 for a chosen $2^\kappa$9.

The mechanism is as follows. Given a model $\lambda \geq \kappa$0 of $\lambda \geq \kappa$1, every extension $\lambda \geq \kappa$2 with $\lambda \geq \kappa$3 contains exactly one new element $\lambda \geq \kappa$4, and the $\lambda \geq \kappa$5-class of $\lambda \geq \kappa$6 must be an infinite $\lambda \geq \kappa$7-class of $\lambda \geq \kappa$8. Conversely, each infinite $\lambda \geq \kappa$9-class $\kappa$0 of $\kappa$1 yields exactly one extension $\kappa$2, and the standard back-and-forth analysis of equivalence relations shows $\kappa$3 iff $\kappa$4. Hence $\kappa$5 equals the number of infinite $\kappa$6-classes of $\kappa$7.

Since for any $\kappa$8 there is a model $\kappa$9 of size $T$0 with exactly $T$1 infinite classes (take $T$2 classes of each finite size, plus $T$3 infinite classes), the paper obtains, for every infinite $T$4:

• some $T$5 of size $T$6 with $T$7 (no infinite classes), and
• $T$8.

The first item alone refutes the conjecture, which predicted $T$9 extensions; the second shows the count can be made exactly $P$0, far below $P$1, for every $P$2. Notably, $P$3 is stable and $P$4 is stably embedded — the $P$5-class of the unique point outside $P$6 is definable over $P$7 by $P$8 — so the counterexample is not an artifact of instability or of pathological interaction between $P$9 and its complement. The theory does, however, fail Hypothesis 2.1(ii) of Shelah–Usvyatsov, and the paper observes that Morleyization can change the induced theory, corroborating Usvyatsov's earlier warning on this point.

The dense counterexample and intermediate values

The second construction uses a dense order. In the language $L$0, take $L$1, choose a point $L$2, and let $L$3; the resulting theory $L$4 requires that the unique point outside $L$5 sit at a nonprincipal cut of the $L$6-part. Then $L$7 fails the Gaifman property, because $L$8 is a model of $L$9 with no nonprincipal cut at which a new element could be inserted. For a base $L$0, extensions correspond exactly to cuts of $L$1, so $L$2, and consequently

$L$3

where $L$4 is the supremum of the number of cuts of a linear order of size at most $L$5 (in the sense of Chernikov–Shelah). Since $L$6, some base $L$7 satisfies $L$8.

The significance of this intermediate value depends on cardinal arithmetic. Mitchell proved that $L$9 is consistent for every $P$00 of uncountable cofinality; for countable cofinality, Chernikov, Kaplan, and Shelah construct a model in which $P$01. Hence, consistently,

$P$02

refuting any repaired version of the conjecture that merely lowers the expected number of extensions below $P$03. Additionally, using the base $P$04 with the lexicographic order, whose branch cuts number exactly $P$05, one obtains for every $P$06 a theory and a base with $P$07.

Dividing lines and an open question

The final section, marked "Future work" and left as comments, raises the question of whether $P$08 can be controlled by a fixed cardinal when $P$09 fails the Gaifman property. The paper's constructions cannot settle this, because both rely on compactness and their behavior varies with the size of the base model. The author notes that a size-independent construction would require a structure capable of coding arbitrary information, and points to Pillay–Shelah's theorem on admissible expansion problems: if the fixed-base extension problem is admissible by finitely many relation symbols, then $P$10 for some $P$11 holds iff there is a uniform finite bound $P$12 with $P$13 for all $P$14. Whether this dichotomy constrains the present counterexamples is left open.

Limitations

Two caveats bear directly on the strength of the results. First, the dense counterexample's refutation of intermediate bounds is conditional: the strict inequality $P$15 holds only under the consistency assumption $P$16, which is known only for specific cofinalities and is not a ZFC theorem. Second, the discrete example, while unconditional, gives $P$17 rather than a bounded value, so it does not answer whether the number of extensions can be uniformly bounded independently of $P$18. Both constructions are also deliberately minimal — one new element per extension — and it is unclear whether theories requiring richer extensions admit similar control.

Conclusion

The paper disproves the Generalized Gaifman Conjecture with two elementary constructions in finite languages: a stable theory with a stably embedded predicate for which $P$19 (and can be $P$20 for some bases), and a dense-order theory realizing $P$21, which is consistently strictly between $P$22 and $P$23. These results show that failure of the Gaifman property imposes essentially no lower bound on the number of relatively non-isomorphic extensions, while leaving open whether any uniform bound is possible.