The role of matroids in model theory

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312Zilber’s ConjectureExamples of uncountably categorical theories:• Infinite set with no additional structure.• Vector space over a given finite (or countable) field.• Projective space over a given finite (or countable) field.• Algebraically closed field of a given characteristic.Conjecture (Zilber)Every uncountably categorical countable first-order theory isessentially of one of the above types.Hrushovski found uncountably many counterexamples.
412 MatroidsDefinitionA matroid is a set M equipped with a finitary closure operatorcl: P(M) → P(M) that satisfies the exchange property.A ⊆ cl AA ⊆ B =⇒ cl A ⊆ cl Bcl(cl A) = cl Acl A = ⋃ {cl F | F ⊆ finite A}a ∈ cl(C ∪ {b}) \ cl C =⇒ b ∈ cl(C ∪ {a}).A basis of a matroid is a minimal generating set / maximalindependent set. Its dimension is the cardinality of any basis.Examples of matroids:• Trivial closure (i.e. cl A = A) on any set.• Linear hull on a vector space. A matrix.• Algebraic closure on an algebraically closed field.• The edges of a graph.
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The role of matroids in model theory

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