Embedding Factorizations for 3-Uniform Hypergraphs II: $r$-Factorizations into $s$-Factorizations

Abstract

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 $r=s=1$.