Kim’s lemmas and tree properties
One of the most important technical steps in the development of simplicity theory in the 1990s was a result now known as Kim’s Lemma: In a simple theory, if a formula ϕ(x;b) divides over a model M, then ϕ(x;b) divides along every Morley sequence in tp(b/M). More recently, variants of Kim’s Lemma have been shown by Chernikov, Kaplan, and Ramsey to follow from, and in fact characterize, two generalizations of simplicity in different directions: the combinatorial dividing lines NTP2 and NSOP1. After surveying the Kim’s Lemmas of the past, I will suggest a new variant of Kim’s Lemma, and a corresponding new model-theoretic tree property, which generalizes both TP2 and SOP1. I will also compare this new tree property with the Antichain Tree Property (ATP), another tree property generalizing both TP2 and SOP1, which was introduced recently by Ahn and Kim. This is joint work with Nick Ramsey.
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