Threshold for stability of weak saturation.

We study the weak Ks ${K}_{s}$‐saturation number of the Erdős–Rényi random graph G(n,p) ${\mathbb{G}}(n,p)$, denoted by wsat(G(n,p),Ks) $\text{wsat}({\mathbb{G}}(n,p),{K}_{s})$, where Ks ${K}_{s}$ is the complete graph on s $s$ vertices. In 2017, Korándi and Sudakov proved that the weak Ks ${K}_{s}$‐saturation number of Kn ${K}_{n}$ is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Korándi and Sudakov. A general upper bound on wsat(G(n,p),Ks) $\text{wsat}({\mathbb{G}}(n,p),{K}_{s})$ is also provided. [ABSTRACT FROM AUTHOR]