$Sz(\cdot)\leqslant ω^ξ$ is rarely a three space property

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

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