Off-diagonal Ramsey numbers for 3-uniform tight hypergraph cycles

Let $C^{(3)}_n$ be the $3$-uniform tight cycle on $n$ vertices, that is, the $3$-uniform hypergraph on a cyclically ordered set of $n$ vertices whose edges are precisely the triples of consecutive vertices in the given cyclic order. We asymptotically determine the off-diagonal Ramsey numbers of $3$-uniform tight cycles. This is an off-diagonal analogue of a result of Haxell, Łuczak, Peng, Rödl, Ruciński and Skokan [Combin.\ Probab.\ Comput.\ 18 (2009), no.\ 1-2, 165--203] on the diagonal Ramsey numbers of $3$-uniform tight cycles.