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An equivalence-relation first-order structure with exactly two infinite classes, one countable and one uncountable, is atomic but lacks finite-tuple homogeneity. Its first-order theory has quantifier elimination: a new point is either assigned a previously used class or an available unused class, and can avoid finitely many forbidden points in that infinite class. Finite equality-and-equivalence patterns therefore isolate all tuple partial types. All elements have the same complete one-type, but no automorphism exchanges the two differently sized classes.

A first-order structure has finite-tuple homogeneity when any two finite tuples of the same parameter-free complete type are related by an automorphism. For countably infinite first-order structures this agrees with extension of finite partial elementary maps to automorphisms. Countable atomic models have this property by a back-and-forth method using isolating logical formulas. Uncountable atomic models can fail it.