On the unconditional subsequence property

We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (fi ) in ( Lpi ) 2, where pi ↓ 1, so that for all ε > 0 and 1 < q 2, every subsequence of (fi ) admits a block basis (1+ε)-equivalent to the Haar basis for Lq . We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so. Published by Elsevier Inc