I recently gave a talk at the *Claremont Colleges Algebra/Number Theory/Combinatorics Seminar* on the topic of this paper, which can be found in my papers page.

For a set let . A function is *regressive* iff for all in with . A set is *min-homogeneous* for iff whenever and .

**Theorem**. For all there exists such that if and is regressive, then there is of size at least and min-homogeneous for .

The theorem (due to Kanamori and McAloon) states a version of the classical Ramsey theorem for regressive functions. We cannot expect to be *homogeneous*, i.e., in general will not be constant. For example, consider . Notice also that without loss of generality , since . It is natural to try to establish the rate of growth of the function that to each assigns the least as in the theorem. Using tools of mathematical logic, as part of a more general result about regressive functions of variables, Kanamori and McAloon showed:

**Theorem**. The function grows faster than any primitive recursive function.

In my paper I show using finite combinatorics methods that grows precisely as fast as Ackermann’s function. This is obtained as part of an analysis of a more general function of two variables, defined as but with the additional requirement that . Obviously, and . The situation for is less clear, although it is of exponential growth.

**Theorem**. We have:

- , , , .
- , so for .
- .

**Question**. Does hold?

Although I have not been able to prove this, I do not expect it to be particularly difficult.