The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{ \H o}s and Rogers introduced a more general problem replacing $K_2$ by $K_s$ for $2\le s<t$. Extending the problem of determining Ramsey numbers they defined the numbers $$ f_{ s,t}(n)=\min \big{ \{ } \max \{ |W| : W\subseteq V(G) \text{ and} G[W] \text{ contains no} K_s\}\big{ \}}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this note, we study an analogous function $f_{ s,t}^{ (3)}(n)$ for 3-uniform hypergraphs. In particular, we show that there are constants $c_1$ and $c_2$ depending only on $s$ such that $$ c_1(\log n)^{ 1/4} \left(\frac{ \log\log n}{ \log\log\log n}\right)^{ 1/2} < f_{ s, s+1}^{ (3)}(n) < c_2 \log n. $$
Categories: math.CO