The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson’s original space
T
∗
T^*
. Every Banach space that is coarsely embeddable into
T
∗
T^*
must be reflexive, and all of its spreading models must be isomorphic to
c
0
c_0
. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson:
T
∗
T^*
coarsely contains neither
c
0
c_0
nor
ℓ
p
\ell _p
for
p
∈
[
1
,
∞
)
p\in [1,\infty )
. We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into
T
∗
T^*
, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to
c
0
c_0
. Also, a purely metric characterization of finite dimensionality is obtained.