  We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable
$\varepsilon$-biorthogonal system but no uncountable $\tau$-biorthogonal system
for $\tau<\varepsilon$. In particular the space have no uncountable
biorthogonal system. We also construct a Banach space $\mathcal X_K$ with an
uncountable $K$-basic sequence but no uncountable $K'$-basic sequence, for
$1\leq K'<K$. A common feature of these examples is that they are both
constructed by recursive amalgamations using a single construction scheme.
