This a specific question about Ramsey type colorings.

The arrow notation: If $\kappa$, $\lambda$, $\mu$ are cardinals and $n<\omega$, then $$\kappa\rightarrow(\lambda)^n_\mu$$ if for any function $f:[\kappa]^n\longrightarrow\mu$ (where $[\kappa]^n$ is the set of subsets of $\kappa$ of size $n$), there is a subset $A\subseteq\kappa$ of size $\lambda$ such that $f$ is constant on $[A]^n$.

I am aware of the traditional Ramsey's Theorem, which talks about this arrow notation for countable cardinals; as well as the Erdős-Rado theorem, which says $$(\beth_n(\kappa))^+\rightarrow(\kappa^+)^{n+1}_{\kappa}$$ for any infinite $\kappa$.

My questions are:

1. For what infinite cardinals $\kappa$ is it true that $\kappa^+\rightarrow(\omega)^2_\kappa$

2. For what infinite cardinals $\kappa$ is it true that $2^\kappa\rightarrow(\omega)^2_\kappa$

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I'm pretty sure the answer is never, but I don't recall the proof off hand. In fact, I think you can replace $\omega$ with 3. –  Shawn Henry Oct 25 '12 at 18:57
@Shawn: Your memory’s correct. –  Brian M. Scott Oct 25 '12 at 19:17

It’s a result of Erdős and Kakutani that $2^\kappa\not\to(3)^2_\kappa$. Let $\Sigma={^\kappa\{0,1\}}$, and define
$$\varphi:[\Sigma]^2\to\kappa:\{\sigma,\tau\}\mapsto\min\{\xi<\kappa:\sigma(\xi)\ne\tau(\xi)\}\;.$$
Let $\rho,\sigma,\tau$ be distinct elements of $\Sigma$; if $\varphi(\rho,\sigma)=\varphi(\sigma,\tau)=\xi$, then $\rho(\xi)=\tau(\xi)$, and therefore $\varphi(\rho,\tau)\ne\xi$.
In particular, $2^\omega\not\to(3)^2_\omega$, and hence $\omega_1\not\to(3)^2_\omega$. It follows that $2^\kappa\not\to(\omega)^2_\kappa$ and $\kappa^+\not\to(\omega)^2_\kappa$ for any infinite cardinal $\kappa$.