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

Abstract

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.