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

$Sz(\cdot)\leqslant \omega^\xi$ is rarely a three space property