On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)n∈N. The relation E0 is Borel reducible to permutative equivalence between normalized block-sequences of (en)n∈N or X is c0 or ℓp saturated for some 1⩽p<+∞. If (en)n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c0 or ℓp, 1