Exact bounds for some hypergraph saturation problems

Let Kp1,...,pdd denote the complete d-uniform d-partite hypergraph with partition classes of sizes p₁, ..., p_d. A hypergraph G⊆Kn,...,nd is said to be weakly Kp1,...,pdd-saturated if one can add the edges of Kn,...,nd\G in some order so that at each step a new copy of Kp1,...,pdd is created. Let W_n(p₁,..., p_d) denote the minimum number of edges in such a hypergraph. The problem of bounding W_n(p₁,..., p_d) was introduced by Balogh, Bollobás, Morris and Riordan who determined it when each p_i is either 1 or some fixed p. In this note we fully determine W_n(p₁,..., p_d). Our proof applies a reduction to a multi-partite version of the Two Families Theorem obtained by Alon. While the reduction is combinatorial, the main idea behind it is algebraic.