We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (f(i)) in (Sigma L-pi)l(2), where p(i) down arrow 1, so that for all epsilon > 0 and 1 < q <= 2, every subsequence of (f(i)) admits a block basis (1 + epsilon)-equivalent to the Haar basis for L-q. We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so. Published by Elsevier Inc.
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