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arXiv · 2507.12631

Stable amalgamation over a predicate and the Gaifman property

Saharon Shelah·Alexander Usvyatsov
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Abstract

We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured that if T is relatively categorical over P, then it has the Gaifman property. We propose a generalized version of this conjecture: if T fails the Gaifman property, then it exhibits non-structure over P, i.e., has many non-isomorphic models over P in many cardinalities. We address this conjecture for countable theories. Motivated by ideas from Classification Theory, we separate this conjecture into two parts: 1) stability over P (a structure property of theories) implies the Gaifman property, and 2) instability over P implies non-structure. In this paper prove the first part of this conjecture. In fact, we prove a stronger statement: an appropriate version of stability implies higher stable amalgamation properties.

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Saharon Shelah, Alexander Usvyatsov. 2025-07-16. Stable amalgamation over a predicate and the Gaifman property. https://arxiv.org/abs/2507.12631

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