Weak saturation numbers of complete bipartite graphs in the clique

The notion of weak saturation was introduced by Bollob\'as in 1968. Let $F$ and $H$ be graphs. A spanning subgraph $G \subseteq F$ is weakly $(F,H)$-saturated if it contains no copy but there exists an ordering $e_1,\ldots,e_t$ $E(F)\setminus E(G)$ such that for each $i \in [t]$, the graph \cup \{e_1,\ldots,e_i\}$ a $H'$ $e_i H'$. Define $wsat(F,H)$ to minimum number edges graph. In this paper, we prove all $t \ge 2$ $n 3t-3$, $wsat(K_n,K_{t,t}) = (t-1)(n + 1 - t/2)$, determine value $wsat(K_n,K_{t-1,t})$ as well. For fixed $2 \le s < t$, also obtain bounds on $wsat(K_n,K_{s,t})$ are asymptotically tight.