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 j = 1i - 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 n≥2, \[ r(F,n) = Ω({n^k}{(log n)2k - 2}).\] In particular, we deduce that the off-diagonal Ramsey number r(F₅,n) is of order n³/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.