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CHAPTER 1 Combinatorial Geometries and Infinitary Logics In this chapter we introduce two of the key concepts that are used throughout the text. In the first section we define the notion of a combinatorial geometry and describe its connection with categoricity in the first order case. The second section establishes the basic notations for infinitary logics. 1.1. Combinatorial Geometries Shelah’s notion of forking provides a notion of independence which greatly generalizes the notion of Van der Waerden [VdW49] by allowing one to study structures with a family of dimensions. (See Chapter II of [Bal88].) But Van der Waerden’s notion is the natural context for studying a single dimension in another guise such a relation is called a combinatorial geometry. Most of this monograph is devoted to structures which are determined by a single dimension in the simplest case, which we study first, the entire universe is the domain of a single combinatorial geometry. One strategy for the analysis of ℵ1-categorical first order theories in a countable vocabulary proceeds in two steps: a) find a definable set which admits a nice dimension theory b) show this set determines the model up to isomorphism. The sufficient condition for a set 𝐷 to have a nice dimension theory is that 𝐷 be strongly minimal or equivalently that algebraic closure on 𝐷 forms a pregeometry in the sense described below. It is natural to attempt to generalize this approach to study categoricity in non-elementary contexts. In this chapter we review the first order case to set the stage. For more detail on this chapter consult e.g. [Bue91]. In the next few chapters, we study the infinitary analog to the first order concept of strong minimality – quasiminimality. In Part 4 we show that the strategy of reducing categoricity to a quasiminimal excellent subset can be carried out for sentences of 𝐿𝜔 1 ,𝜔 . Definition 1.1.1. A pregeometry is a set 𝐺 together with a dependence relation 𝑐𝑙 : 𝒫(𝐺) → 𝒫(𝐺) satisfying the following axioms. A1. 𝑐𝑙(𝑋) = ∪ {𝑐𝑙(𝑋′): 𝑋′ ⊆𝑓𝑖𝑛 𝑋} A2. 𝑋 ⊆ 𝑐𝑙(𝑋) A3. 𝑐𝑙(𝑐𝑙(𝑋)) = 𝑐𝑙(𝑋) A4. (Exchange) If 𝑎 ∈ 𝑐𝑙(𝑋𝑏) and 𝑎 ∕ ∈ 𝑐𝑙(𝑋), then 𝑏 ∈ 𝑐𝑙(𝑋𝑎). If points are closed the structure is called a geometry. Definition 1.1.2. A geometry is homogeneous if for any closed 𝑋 ⊆ 𝐺 and 𝑎,𝑏 ∈ 𝐺 − 𝑋 there is a permutation of 𝐺 which preserves the closure relation (i.e. an automorphism of the geometry), fixes 𝑋 pointwise, and takes 𝑎 to 𝑏. 3 http://dx.doi.org/10.1090/ulect/050/01