Abstract. We present simple constructions of trees and gaps using a general construction scheme that can be useful in constructing many other structures. As a result, we solve a natural problem about Hausdorff gaps in the quotient algebra P(ω)/Fin found in the literature. As it is well known Hausdorff gaps can sometimes be filled in ω 1 -preserving forcing extensions. There are two natural conditions on Hausdorff gaps, dubbed S and T in the literature, that guarantee the existence of such forcing extensions. In part, these conditions are motivated by analogies between fillable Hausdorff gaps and Souslin trees. While the condition S is equivalent to the existence of ω 1 -preserving forcing extensions that fill the gap, we show here that its natural strengthening T is in fact strictly stronger.

Abstract. We construct a Banach space X ε with an uncountable ε-biorthogonal system but no uncountable τ -biorthogonal system for τ < ε(1 + ε) −1 . In particular the space have no uncountable biorthogonal system. We also construct a Banach space X K with an uncountable K-basic sequence but no uncountable K -basic sequence, for 1 ≤ K < K. A common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme.

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