Banach spaces from a construction scheme

We construct a Banach space XεXε with an uncountable ε-biorthogonal system but no uncountable τ  -biorthogonal system for τ<ε(1+ε)−1τ<ε(1+ε)−1. In particular the space has no uncountable biorthogonal system. We also construct a Banach space XKXK with an uncountable K  -basic sequence but no uncountable K′K′-basic sequence, for 1≤K′