The independent neighborhoods process

A triangle T (r) in an r-uniform hypergraph is a set of r + 1 edges such that r of them share a common (r − 1)-set of vertices and the last edge contains the remaining vertex from each of the first r edges. Our main result is that the random greedy triangle-free process on n points terminates in an r-uniform hypergraph with independence number O((n log n)). As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number r(T ,K (r) s ) has order of magnitude s/ log s. This answers questions posed in [4, 10] and generalizes the celebrated results of Ajtai-Komlós-Szemerédi [1] and Kim [9] to hypergraphs.