Let f:Z_{+}→R be an increasing function. We say that an infinite word w is abelian f(n)-saturated if each factor of length n contains Θ(f(n)) abelian nonequivalent factors. We show that binary infinite words cannot be abelian n^{2}-saturated, but, for any ε>0, they can be abelian n^{2−ε}-saturated. There is also a sequence of finite words (w_{n}), with |w_{n}|=n, such that each w_{n} contains at least Cn^{2} abelian nonequivalent factors for some constant C>0. We also consider saturated words and their connection to palindromic richness in the case of equality and k-abelian equivalence.

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