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Let $\lambda\geq \omega_2$ be a regular cardinal. The weak reflection principle for $[\lambda]^\omega$ ($WRP([\lambda]^\omega)$) asserts that for any stationary $S\subset [\lambda]^\omega$ there exists $\omega_1\subset W\in [\lambda]^{\aleph_1}$ such that $S\cap [W]^\omega$ is stationary.

Adding one more cardinal parameter, we can talk about $WRP([\lambda]^\omega, <\kappa)$, which asserts for any stationary $S\subset [\lambda]^\omega$ there exists $\omega_1\subset W\in [\lambda]^{<\kappa}$ such that $S\cap [W]^\omega$ is stationary. So $WRP([\lambda]^\omega)\equiv WRP([\lambda]^\omega, <\aleph_2)$

Is the following known:

Suppose $\neg WRP([\aleph_{n+1}]^\omega, <\aleph_{n})$ for all $n\in \omega, n\geq 2$, can we conclude $\neg WRP([\aleph_{\omega+1}]^\omega, <\aleph_\omega)$?

Note with the hypothesis, necessarily, $WRP([\aleph_{\omega+1}]^\omega, <\aleph_n)$ fails for all $n\in \omega$.

The motivation is that if we talk about the ``ordinary'' stationary reflection on $\lambda\cap cof(\omega)$, then the answer is negative.

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