The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C-n((3)) denote the 3-uniform tight cycle, that is, the hypergraph with vertices v(1),...,v(n) and edges v(1)v(2)v(3), v(2)v(3)v(4), ... ,v(n-1)v(n)v(1), v(n)v(1)v(2). We prove that the smallest integer N = N(n) for which every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C-n((3)) is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rodl.