Pairwise balanced designs with prescribed minimum dimension

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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.