  For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding
to $s$ distinct points on a circle and edge set given by the $s$ cyclic
intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$
(mod 3) we prove that there are positive constants $a$ and $b$ with
$$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a
probabilistic construction. The upper bound for $s>5$ is proved by using
supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for
$s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we
develop.
