This is an announcement for the paper “$Sz(\cdot)\leqslant ω^ξ$ is rarely a three space property” by R.M. Causeyhttps://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M.

Abstract: We prove that for any non-zero, countable ordinal $\xi$ which is not additively indecomposable, the property of having Szlenk index not exceeding $\omega^\xi$ is not a three space property. This complements a result of Brooker and Lancien, which states that if $\xi$ is additively indecomposable, then having Szlenk index not exceeding $\omega^\xi$ is a three space property.

https://arxiv.org/abs/1912.13429