A consistency theorem for cardinal sequences of length $<{\omega }_3$

Abstract

We prove that if $\lambda$ is a fixed uncountable cardinal and $f = \langle \ka_{\al} : \al < δ\rangle$ is a sequence of infinite cardinals where $\delta <{\omega }_3$ and $\ka_{\al}\in \{\om,λ\}$ for each $\al < δ$ in such a way that $f^{-1}\{\om\}$ is $\om_2$-closed in $\delta$, then it is consistent that there is a scattered Boolean space whose cardinal sequence is $f$.

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