The Threshold for Ackermannian Ramsey Numbers

For a function g : N → N, the g-regressive Ramsey number of k is the least N so that N min −→ (k)g. This symbol means: for every c : [N ] → N that satisfies c(m, n) ≤ g(min{m, n}) there is a min-homogeneous H ⊆ N of size k, that is, the color c(m, n) of a pair {m, n} ⊆ H depends only on min{m, n}. It is known ([4, 5]) that Id-regressive Ramsey numbers grow in k as fast as Ack(k), Ackermann’s function in k. On the other hand, for constant g, the g-regressive Ramsey numbers grow exponentially in k, and are therefore primitive recursive in k. We compute below the threshold in which g-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian, by proving: Theorem. Suppose g : N → N is weakly increasing. Then the g-regressive Ramsey numbers are primitive recursive if an only if for every t > 0 there is some Mt so that for all n ≥ Mt it holds that g(m) < n and Mt is bounded by a primitive recursive function in t.