The Ramsey number for hypergraph cycles I

Let C n denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v 1 , … , v n and edges v 1 v 2 v 3 , v 3 v 4 v 5 , v 5 v 6 v 7 , … , v n - 1 v n v 1 . We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C n , where N is asymptotically equal to 5 n / 4 . Moreover this result is (asymptotically) best possible.