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**1 - 4**of**4**### Vapnik-Chervonenkis density in some theories without the independence property

, 2011

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### MORLEY RANK IN HOMOGENEOUS MODELS

"... Abstract. We define an appropriate analog of the Morley rank in a totally transcendental homogeneous model with type diagram D. We show that if RM[p] = α, then for some 1 ≤ n < ω the type p has n, but not n + 1, distinct D-extensions of rank α. This is surprising, because the proof of the statem ..."

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Abstract. We define an appropriate analog of the Morley rank in a totally transcendental homogeneous model with type diagram D. We show that if RM[p] = α, then for some 1 ≤ n < ω the type p has n, but not n + 1, distinct D-extensions of rank α. This is surprising, because the proof of the statement in the first-order case depends heavily on compactness. We also show that types over (D, ℵ0)-homogeneous models have multiplicity (Morley degree) 1.