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on Erdős #1173 · set theory / combinatorics

The statement

Assume the generalised continuum hypothesis. Let

f: \omega_{\omega+1}\to [\omega_{\omega+1}]^{\leq \aleph_\omega}

be a set mapping such that

\lvert f(\alpha)\cap f(\beta)\rvert <\aleph_\omega

for all $\alpha\neq \beta$ . Does there exist a free set of cardinality $\aleph_{\omega+1}$ ?

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