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 . Erdős and Rogers introduced a more general problem replacing  K 2 by  K s for 2 ≤ s < t . Extending the problem of determining Ramsey numbers they defined the numbersf s , t( n ) = min { max { | W | : W ⊆ V ( G )andG [ W ]contains noK s } } ,where the minimum is taken over all K t ‐free graphs  G of order  n . In this note, we study an analogous functionf 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 thatc 1( log n ) 1 / 4log log n log log log n1 / 2 < f s , s + 1 ( 3 )( n ) < c 2 log n .