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 ƒ : [m; l]ƒ : [m; l]→ ℕ admits a min-homogeneous set of size n. Analysis of this function also leads to the simplest known proof that the regressive Ramsey numbers have rate of growth at least Ackermannian. Together, these results give a purely combinatorial proof that, for each m; g(•,m) has 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 value under g improving the ones provided by our general argument.
Caicedo, Andrés Eduardo. (2010). "Regressive Functions on Pairs". European Journal of Combinatorics, 31(3), 803-812. http://dx.doi.org/10.1016/j.ejc.2009.07.010