Regressive functions on pairs

We compute an explicit upper bound for the regressive Ramsey numbers by a combinatorial argument, the corresponding function being of Ackermannian growth. For this, we look at the more general problem of bounding g(n,m)g(n,m), the least ll such that any regressive function f:[m,l][2]→Nf:[m,l][2]→N admits a min-homogeneous set of size nn. An analysis of this function also leads to the simplest known proof that the regressive Ramsey numbers have a rate of growth at least Ackermannian. Together, these results give a purely combinatorial proof that, for each m,g(⋅,m)m,g(⋅,m) has a rate of growth precisely Ackermannian, considerably improve the previously known bounds on the size of regressive Ramsey numbers, and provide the right rate of growth of the levels of gg. For small numbers we also find bounds on their values under gg improving those provided by our general argument.