Albiac Alesanco, Fernando JoséAnsorena, José L.Dilworth, S. J.Kutzarova, Denka2019-12-202021-03-1520190022-123610.1016/j.jfa.2018.08.015https://academica-e.unavarra.es/handle/2454/35933It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k(m)[B](m=1)(infinity) of its conditionality constants verifies the estimate k(m)[B] = O(log m) and that if the reverse inequality log m =O(k(m)[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k(m)[B] =O(log m)(1-epsilon) for some epsilon > 0. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k(m)[B] = O(log m) and superreflexiye classical Banach spaces having for every epsilon > 0 quasi-greedy bases B with k(m)[B] = O(log m)(1-epsilon). Moreover, in most cases those bases will be almost greedy.33 p.application/pdfeng© 2018 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0.Conditionality constantsQuasi-greedy basisAlmost greedy basisSubsymmetric basisinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccess