We generalize a result of Shelah that draws positive Ramsey-theoretic conclusions (the existence of infinite free sequences in algebras) from a “drastic” failure of the Singular Cardinal Hypothesis at $\aleph_\omega$. We show that the connection between these two apparently unrelated phenomena is topological and lifts to more general settings: Shelah’s theorem does not really need all of the machinery associated with pcf theory at $\aleph_\omega$. This allows us to obtain stronger results, and uncovers a dichotomy that may have further applications.