Finite-pattern definability conjecture for invariant types

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Let p(x)p(x) be a global MM-invariant type and let Δ\Delta be a finite set of formulas.

Finite-pattern definability conjecture. There is a finite set Δ′\Delta' of formulas such that, for every ϕ(x;y)∈p\phi(x;y)\in p and b,b′∈Ub,b'\in\mathcal U, if

tp⁡Δ′(b/M)=tp⁡Δ′(b′/M),\operatorname{tp}_{\Delta'}(b/M)=\operatorname{tp}_{\Delta'}(b'/M),

then

p⊢ϕ(x;b)↔ϕ(x;b′).p\vdash \phi(x;b)\leftrightarrow\phi(x;b').

The paper states that a positive answer to the invariant-type quotient conjecture would imply this property. Its status is not resolved in the supplied text.

References

Primary source

Pierre Simon, “Dp-minimality: invariant types and dp-rank”, arXiv:1210.4479 (2014).

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