Sharp thresholds for hypergraph regressive Ramsey numbers

The f -regressive Ramsey number R f (d, n) is the minimum N such that every colouring of the d-tuples of an N -element set mapping each x1, . . . , xd to a colour ≤ f(x1) contains a min-homogeneous set of size n, where a set is called min-homogeneous if every two d-tuples from this set that have the same smallest element get the same colour. If f is the identity, then we are dealing with the standard regressive Ramsey numbers as defined by Kanamori and McAloon. In this paper we classifiy the growth-rate of the regressive Ramsey numbers for hypergraphs in dependence of the growth-rate of the parameter function f . The growth-rate has to be measured against the scale of fast-growing Hardy functions Fα indexed by towers of exponentiation in base ω. Our results give a sharp classification of the thresholds at which the f -regressive Ramsey numbers undergoe a drastical change in growth-rate. The case of graphs has been treated of Lee, Kojman, Omri and Weiermann. We extend their results to hypergraphs of arbitrary dimension. From the point of view of logic, our results classify the provability of the Regressive Ramsey Theorem for hypergraphs of fixed dimension with respect to the subsystems of Peano Arithmetic with restricted induction principles.