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 $\aleph_{\omega+1}$?