Joint work with Wee Kee Tang We consider the problem of determining the existence of higher order ` 1 -spreading models in mixed Tsirelson spaces. Let (θn) be a nonincreasing null sequence in (0, 1) such that θm+n ≥ θmθn for all m, n. Argyros, Deliyanni and Manoussakis proved that if lim θ 1/n n = 1, then every block subspace of T[(θn, Sn)∞

n=1] contains an `1-Sω-spreading model. We have the following results. 1. The mixed Tsirelson space T[(θn, Sn) ∞ n=1] contains an ` 1 -Sω- spreading model if and only if limn lim supm θm+n/θm > 0.

2. Let X be a block subspace of the mixed Tsirelson space T[(θn, Sn) ∞ n=1]. The following are equivalent. (a) The Bourgain `1 -index of X is ω ω·2. (b) X contains an `1-Sω- spreading model. (c) Every normalized block sequence in X has a further normalized block (xn) that is equivalent to the sequence (ekn). Here (ek) is the unit vector basis of T[(θn, Sn) ∞ n=1] and kn = max supp xn. Moreover, if every block subspace X has the equivalent properties above, then T[(θn, Sn) ∞ n=1] is arbitrarily distortable. Note that the result of Argyros et al. can be deduced from (2).