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 hyper-edges of the complete k-uniform hypergraph K^k _n on n vertices. Start with the empty hypergraph on n vertices. Successively consider the hyperedges e of K^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.