The stabilized set of p's in Krivine's theorem can be disconnected

For any closed subset F   of [1,∞][1,∞] which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space X with a 1-unconditional basis is constructed so that in each block subspace Y of X  , ℓpℓp is finitely block represented in Y   if and only if p∈Fp∈F. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace Y of X there is a dense subset G of F such that the spreading models admitted by Y   are exactly the ℓpℓp for p∈Gp∈G.