  The dimension of a linear space is the maximum positive integer $d$ such that
any $d$ of its points generate a proper subspace. For a set $K$ of integers at
least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space
on $v$ points whose lines (or blocks) have sizes belonging to $K$. We show
that, for any prescribed set of sizes $K$ and lower bound $d$ on the dimension,
there exists a PBD$(v,K)$ of dimension at least $d$ for all sufficiently large
and numerically admissible $v$.
