Sharp threshold for $K_4$-percolation

Abstract

We locate the critical threshold $p_c$ at which it becomes likely that the complete graph $K_n$ can be obtained from the Erd\H{o}s-R\'enyi graph ${\cal G}_{n,p}$ by iteratively completing copies of $K_4$ minus an edge. This refines work of Balogh, Bollob\'as and Morris that bounds the threshold up to multiplicative constants.