  We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds
(without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d
n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum
possible growth. Moreover, the constant $c_d$ is optimal modulo a singly
exponential factor in $d$. The pure simplicial complexes improve on a
construction of the second author that achieved $c_d n^{2d/3}$. For
pseudo-manifolds without boundary, as far as we know, no construction with
diameter greater than $n^2$ was previously known.
