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  1. Model theory and combinatorics of banned sequences.Hunter Chase & James Freitag - 2022 - Journal of Symbolic Logic 87 (1):1-20.
    We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill [4].
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  • Thicket density.Siddharth Bhaskar - 2021 - Journal of Symbolic Logic 86 (1):110-127.
    We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.
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  • The Dividing Line Methodology: Model Theory Motivating Set Theory.John T. Baldwin - 2021 - Theoria 87 (2):361-393.
    We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...)
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