Stable Forking Conjecture

• Stable theory: a complete theory $T$ such that no formula has the order property — no $\varphi(x,y)$ and tuples $(a_i)_{i<\omega}, (b_i)_{i<\omega}$ with $\models \varphi(a_i, b_j) \iff i < j$.
• Forking: a syntactic/semantic independence notion generalizing linear independence and algebraic independence; $p$ forks over $A$ if it implies a finite disjunction of formulas each of which divides over $A$.
• Stable formula: a formula $\varphi(x,y)$ that does not have the order property, even though the ambient theory $T$ might not itself be stable.

• The conjecture is known to hold in several restricted settings, including theories of finite Morley rank and certain classes of superstable theories.
• It is known to be closely tied to structural questions about the geometry of forking (e.g. one-basedness and the presence of definable group/field structure), and much of the partial progress comes from analyzing these special geometric configurations rather than a fully general argument.
• No counterexample is known within stable theories; the conjecture remains open in general.

This is a foundational structural question about stable theories, tying together the syntactic notion of forking with the more geometric/definable-set-theoretic notion of stability of a formula. The exact original source is not settled here — if you know the precise citation, please suggest an edit.

• Pillay, A. "Geometric Stability Theory." Oxford University Press, 1996 (background on forking, stability, and related conjectures).
• Various survey articles and lecture notes in stability theory discuss the conjecture and its partial cases; readers should consult recent model theory surveys for up-to-date status.