Partition calculus question For $m,n,k < \omega$, consider the equation
$X \to (\omega \times k)^{m}_{n}$
What is the smallest $X$ known to satisfy it?
Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but it fails badly for $m>2$, and actually I hardly managed to find anything in the literature concerning $m>2$. 
The only bound I know is the one given by Erdos-Rado - $\beth_{\omega}$. So $\aleph_{\omega}$ works assuming $GCH$. Can it be proved to work in $ZFC$? Any improvements and/or references are very welcome.
Also I would be happy to hear any results for the same equation with $\omega$ replaced by an arbitrary ordinal $\lambda$.
 A: Here is what I know about this (and mostly learned from my student Thilo Weinert):
It is known that $\mathbb Q\not\rightarrow(\omega+1,4)^3$.
I.e., there is a coloring of the unordered triples of rationals with two colors $0$ and $1$
such that no set of rationals of ordertype $\omega+1$ is homogeneous of color $0$ and
no set of size 4 is homogeneous of color 1.  
On the other hand, Milner and Prikry showed $\omega_1\rightarrow(\omega\cdot 2,4)^3$
and Jones proved $\omega_1\to (\omega+m,n)^3$ for all $n,m\in\omega$.
Milner and Prikry conjectured that $\omega_1\rightarrow(\alpha,n)^3$ holds for all
$n\in\omega$ and all countable $\alpha$.  
Note that all the positive results are unsymmetric, i.e., they only give finite homogeneous sets in one of the colors.
Also, there is a definable (continuous, actually) coloring of the unordered triples of $2^\omega$ (ordered lexicographically) with two colors such that
every infinite homogeneous set has ordertype either $\omega$ or $\omega^*$.  
Finally, let us consider this coloring:  for each $\alpha\lneq\omega_1$ fix a wellordering $\leq_\alpha$ of $\alpha$ of ordertype $\leq\omega$.
Given three ordinals $\alpha\lneq\beta\lneq\gamma\lneq\omega_1$, let $c(\alpha,\beta,\gamma)=0$ if $\leq_\alpha$ agrees with the usual ordering of ordinals
on $\{\alpha,\beta\}$ and otherwise let $c(\alpha,\beta,\gamma)=1$.
I think there is no homogeneous set of color $0$ of ordertype $\omega+2$ and no
homogeneous set of color $1$ of ordertype $\omega+1$.  This would show $\omega_1\not\rightarrow(\omega+2)^3_2$.
A: I think that the general conjecture $\omega_1\to(\alpha,n)^3$ goes back to Erdos and Rado. 
