Hypergraph Ramsey numbers: tight cycles versus cliques

For $s\ge 4$, the 3-uniform tight cycle $C_s^3$ 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 $s\equiv ̸0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C_s^3,K_t^3)<2^{b{t}^2\log t}\mathrm{.}$$ 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.