Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness.#R##N##R##N# further investigate categoricity transfers in tame AECs.  use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals:#R##N##R##N##R##N##R##N#Theorem 0.1. #R##N#LetKbe an AEC and let  LS(K)≤λ<θLS(K)≤λ<θbe cardinals. IfKhas a type-full good  [λ,θ][λ,θ]-frame andKis categorical in both λ and  θ+θ+, thenKis categorical in all  μ∈[λ,θ]μ∈[λ,θ].#R##N##R##N##R##N##R##N##R##N# deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶℶ(2LS(K))+ℶℶ(2LS(K))+ to ℶ(2LS(K))+ℶ(2LS(K))+ assuming that the AEC is LS(K)LS(K)-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M∪{a}M∪{a} or (using an unpublished claim of Shelah) that the ak generalized continuum hypothesis holds.
['Sebastien Vasey']
I am writing a research paper on a idea Downward categoricity from a successor inside a good frame
[{"from": "human", "value": "I am writing a research paper on a idea Downward categoricity from a successor inside a good frame"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness.#R##N##R##N# further investigate categoricity transfers in tame AECs.  use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals:#R##N##R##N##R##N##R##N#Theorem 0.1. #R##N#LetKbe an AEC and let \u00a0LS(K)\u2264\u03bb<\u03b8LS(K)\u2264\u03bb<\u03b8be cardinals. IfKhas a type-full good \u00a0[\u03bb,\u03b8][\u03bb,\u03b8]-frame andKis categorical in both \u03bb and \u00a0\u03b8+\u03b8+, thenKis categorical in all \u00a0\u03bc\u2208[\u03bb,\u03b8]\u03bc\u2208[\u03bb,\u03b8].#R##N##R##N##R##N##R##N##R##N# deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from \u2136\u2136(2LS(K))+\u2136\u2136(2LS(K))+ to \u2136(2LS(K))+\u2136(2LS(K))+ assuming that the AEC is LS(K)LS(K)-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M\u222a{a}M\u222a{a} or (using an unpublished claim of Shelah) that the ak generalized continuum hypothesis holds."}]