In the language of partition calculus, I want to know if the following is true:
Let $n$ be an integer, $n \geq 2$. Then $\aleph_n \rightarrow (\aleph_0)^2_{\aleph_0}$.
I'm new to infinitary combinatorics, and it's possible that I used the wrong notation, so I will explain with words. Let $G$ be a graph such that the number of vertices is $\aleph_n$, $n \geq 2$. We color each edge of $G$ with countably many colors. Then I can find a countably infinite set $S$ of vertices, and each edge between two vertices in $S$ will be the same color.
The reason I require $n \geq 2$ is that the interval of real numbers $[0,1]$ has cardinality $|[0,1]| \geq \aleph_1$, but if I color each edge between real numbers in the interval using the rule
$\{a,b\}$ has the color $k \in \bf{Z^{\geq 0}}$ if and only if $|a-b| \in (\big (\dfrac{1}{2}\big)^{k+1}, \big(\dfrac{1}{2} \big )^k ]$
then I can color each edge with countably many colors while avoiding monochromatic triangles. So I can demonstrate that my initial statement is false for $n = 1$. However, I am not sure about graphs with higher cardinalities of vertices, and I am unable to find a result in the most commonly referenced works about partition calculus. I'd like to know if this is a known result and where I can find a proof.
I'm also curious about the following kinds of statements, and any help would be appreciated.
$\aleph_3 \rightarrow (\aleph_0)^2_{\aleph_1}$
$\aleph_{n+2} \rightarrow (\aleph_0)^2_{\aleph_n}$
$\aleph_{n+2} \rightarrow (\aleph_n)^2_{\aleph_n}$
$\kappa \rightarrow (\aleph_0)^2_{\lambda}$ where $\kappa$ is an inaccessible cardinal and $\lambda$ is an accessible cardinal.
Thank you.