Edit Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell_1$. See the paper > Odell, E.; Schlumprecht, Th. *A problem on spreading models*. J. Funct. Anal. 153 (1998), no. 2, 249--261. [MR1614578](http://www.ams.org/mathscinet-getitem?mr=1614578) So $\beta =2$ and yet reflexive. Ignore my first answer below. If you allow renormings of the space when computing $\beta$ then $\beta(S_X)=2$ implies that $X$ is non-reflexive. This follows (I think) from a [theorem of Odell and Schlumprecht.][1] [1]: http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00251-3/S0894-0347-98-00251-3.pdf