A note on the random greedy independent set algorithm

Abstract: Let $r\ge 3$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D > N^\varepsilon $ for some $ \varepsilon>0 $. Consider the random greedy algorithm for forming an independent set in $ \mathcal{H}$. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices $v$ with the property that $v$ is not in the current independent set $I$ and $ I \cup {v}$ contains no edge in $ \mathcal{H}$). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of $ \mathcal{H} $; that is, the process terminates at a maximal independent set. We prove that if $ \mathcal{H}$ satisfies certain degree and codegree conditions then there are $ \Omega\left( N \cdot ( (\log N) / D )^{\frac{1}{r-1}} \right) $ vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the $ H$-free process due to Bohman and Keevash and produces objects of interest in additive combinatorics.