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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

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.

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

So $\beta =2$ and yet reflexivity holds.

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.

Edit

Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell_1$. See the paper of Odell and Schlumprecht in JFA titled A problem about spreading models. So $\beta =2$ and yet reflexive.

Ignore the below my first answer.

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.

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

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.

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.

Edit

Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell_1$. See the paper of Odell and Schlumprecht in JFA titled A problem about spreading models. So $\beta =2$ and yet reflexive.

Ignore the below my first answer.

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.