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 ) , the least l such that any regressive function f : [ m , l ] [ 2 ] → N admits a min-homogeneous set of size n . 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 ) 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 g . For small numbers we also find bounds on their values under g improving those provided by our general argument.