Hypergraph Ramsey numbers of cliques versus stars (original) (raw)

Let K (3) m denote the complete 3-uniform hypergraph on m vertices and S (3) n 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 (3) 4 , S (3) n) exhibits an unusual intermediate growth rate, namely, 2 c log 2 n ≤ r(K (3) 4 , S (3) n) ≤ 2 c ′ n 2/3 log n for some positive constants c and c ′. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum N such that any 2-edge-coloring of the Cartesian product KN KN contains either a red rectangle or a blue Kn?