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 ![X^{[2]}=\{(n,m)\in X^2:n<m\}](http://l.wordpress.com/latex.php?latex=X%5E%7B%5B2%5D%7D%3D%5C%7B%28n%2Cm%29%5Cin+X%5E2%3An%3Cm%5C%7D&bg=ffffff&fg=000000&s=0 "X^{[2]}=\{(n,m)\in X^2:n<m\}")
. A function ![f:X^{[2]}\to{\mathbb N}](http://l.wordpress.com/latex.php?latex=f%3AX%5E%7B%5B2%5D%7D%5Cto%7B%5Cmathbb+N%7D&bg=ffffff&fg=000000&s=0 "f:X^{[2]}\to{\mathbb N}")
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 ![f\upharpoonright H^{[2]}](http://l.wordpress.com/latex.php?latex=f%5Cupharpoonright+H%5E%7B%5B2%5D%7D&bg=ffffff&fg=000000&s=0 "f\upharpoonright H^{[2]}")
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.
