We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula $$\phi (x,y)$$, the (...) following are equivalent. (shrink)

We give a new characterization of _SOP_ (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has _OP_ (the order property) if and only if it has _IP_ (the independence property) or _SOP_, in several ways by characterizing various notions in functional analytic style. We point out some connections (...) between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories. (shrink)

This paper has three parts. First, we establish some of the basic model theoretic facts about, the Tsirelson space of Figiel and Johnson [20]. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model‐theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) has the non independence property (NIP); (2) every Banach space that is ℵ0‐categorical up to small perturbations embeds c0 or () almost isometrically; (...) consequently the (continuous) first‐order theory of does not characterize, up to almost isometric isomorphism. (shrink)