On Generalized Ramsey Numbers for 3-Uniform Hypergraphs

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. Erdos and Rogers introduced a more general problem replacing K-2 by Ks for 2 <= s <t. Extending the problem of determining Ramsey numbers they defined the numbers f(s,t)(n) = min{max{|W|:W subset of V(G) and G[W] contains no K-s}},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 c1(log n)(1/4) (loglogn/logloglogn)(1/2) < f(s,s+1)((3))(n) < c(2) log n.