Threshold for stability of weak saturation

Abstract

We study the weak $K_s$-saturation number of the Erdős–Rényi random graph $G\left(n,p\right)$, denoted by $\text{wsat}\left(G\left(n,p\right),K_s\right)$, where $K_s$ is the complete graph on $s$ vertices. In 2017, Korándi and Sudakov proved that the weak $K_s$-saturation number of $K_n$ 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 $\text{wsat}\left(G\left(n,p\right),K_s\right)$ is also provided.