On the random greedy F-free hypergraph process

Let F be a strictly k-balanced k-uniform hypergraph with e(F ) ≥ |F | − k + 1 and maximum co-degree at least two. The random greedy F -free process constructs a maximal F -free hypergraph as follows. Consider a random ordering of the hyperedges of the complete k-uniform hypergraph K n on n vertices. Start with the empty hypergraph on n vertices. Successively consider the hyperedges e of K n in the given ordering, and add e to the existing hypergraph provided that e does not create a copy of F . We show that asymptotically almost surely this process terminates at a hypergraph with Õ(nk−(|F |−k)/(e(F )−1)) hyperedges. This is best possible up to logarithmic factors.