Diamond on ladder systems and countably metacompact topological spaces

Joint work with Rodrigo Rey Carvalho and Tanmay Inamdar.

Abstract. Leiderman and Szeptycki proved that a single Cohen real introduces a ladder system $L$ over $\aleph_1$ for which the space $X_L$ is not a $\Delta$-space. They asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\Delta$-space.  We prove that an affirmative answer holds for $\kappa=cf(\beth_{\omega+1})$. This is an application of a theorem of Shelah concerning diamond on ladder systems, and we include a streamlined presentation of this result. Assuming $\beth_\omega=\aleph_\omega$, we get an example at a much lower cardinal, namely $\kappa=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $\omega$-bounded.

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