Hypergraph Ramsey numbers of cliques versus stars

Let K-m((3)) denote the complete 3-uniform hypergraph on m vertices and S-n((3)) the 3-uniform hypergraph on n + 1 vertices consisting of all ( (n) (2)) edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number r(K-4((3)), S-n((3))) exhibits an unusual intermediate growth rate, namely, [GRAPHICS] . for some positive constants c and c'. The proof of these bounds brings in a novel Ramsey problem on grid graphs whichmay be of independent interest: what is the minimum N such that any 2-edge-coloring of the Cartesian product K-N square K-N contains either a red rectangle or a blue K-n?