Percentages in pairwise balanced designs

Let K = {k₁,...,k_m} be a set of block sizes, and let {p₁,...,p_m} be nonnegative numbers with σ^m_i=1p_i = 1. We prove the following theorem: for any ε{lunate} > 0, if a (v,K,1) pairwise balanced design exists and v is sufficiently large, then a (v,K,1) pairwise balanced design exists in which the fraction of pairs appearing in blocks of size k_i is p_i±ε{lunate} for every i. We also show that the necessary conditions for a pairwise balanced design having precisely the fraction p_i of its pairs in blocks of size k_i for each i are asymptotically sufficient.