Off-diagonal Ramsey numbers for slowly growing hypergraphs

For a k-uniform hypergraph F and a positive integer n, the Ramsey number r (F, n) denotes the minimum N such that every N-vertex F-free k-uniform hypergraph contains an independent set of n vertices. A hypergraph is slowly growing if there is an ordering e₁, e₂, , eₜ of its edges such that |eᵢ ₉ = ₁^i - 1eⱼ| 1 for each i \2, , t\. We prove that if k 3 is fixed and F is any non k-partite slowly growing k-uniform hypergraph, then for n2, \ r (F, n) = (nᵏ (n) ^{2k - 2}). \ In particular, we deduce that the off-diagonal Ramsey number r (F₅, n) is of order n^3/polylog (n), where F₅ is the triple system \123, 124, 345\. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.