Embedding factorizations for 3-uniform hypergraphs II: r-factorizations into s-factorizations

Motivated by a 40-year-old problem due to Peter Cameron on extending partial parallelisms, we provide necessary and sufficient conditions under which one can extend an r-factorization of a complete 3-uniform hypergraph on m vertices, K-m(3), to an s-factorization of K-n(3). This generalizes an existing result of Baranyai and Brouwer where they proved it for the case r = s = 1.