A Counterexample to the Stable Forking Conjecture

The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \not\downarrow_C b$ then there is $\varphi(x,\bar b) \in \mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\varphi(x,y)$ is stable. The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.

The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \not\downarrow_C b$ then there is $\varphi(x,\bar b) \in \mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\varphi(x,y)$ is stable. The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.