The \(Q_2\)-free process in the hypercube (Q2213808)

Summary: The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the \(Q_2\)-free process in \(Q_d\) and the random subgraph of \(Q_d\) it generates. Our main result is that with high probability the graph resulting from this process has at least \(cd^{2/3} 2^d\) edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.