Weak saturation of multipartite hypergraphs (Q6143930)

Let \(F\) and \(H\) be \(q\)-uniform hypergraphs. A subgraph \(G \subseteq F\) is \textit{weakly \(H\)-saturated} in \(F\) if the edges of \(F \setminus G\) can be ordered as \(e_1, \dots , e_k\) such that for each \(i \in [k]\), the hypergraph \(G \cup {e_1, \dots , e_i}\) contains an isomorphic copy of \(H\) containing the edge \(e_i\). The \textit{weak saturation number} of \(H\) in \(F\), denoted wsat\((F,H)\), is the minimum number of edges in a weakly \(H\)-saturated subgraph of \(F\).\par The main contribution of the paper is the exact determination of the weak saturation number of complete multipartite \(q\)-graphs in the directed setting, generalizing a theorem by \textit{N. Alon} [J. Comb. Theory, Ser. A 40, 82--89 (1985; Zbl 0578.05002)]. Additionally, the authors establish a link between weak saturation numbers of bipartite graphs in the clique and in a complete bipartite host graph, thereby answerign a question posed by \textit{G. Kronenberg} et al. [J. Comb. Theory, Ser. A 178, Article ID 105357, 16 p. (2021; Zbl 1457.05089)].