Partition relation, almost a Ramsey cardinal?

Question

Is it consistent with ZFC to have a cardinal $\kappa$ which is not Ramsey and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds for some $n\in \omega$?

The partition relation $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds iff for every function $f:[\kappa]^{<\omega}\rightarrow \omega$, there is an $A\in [\kappa]^\kappa$ such that for each $l\in \omega$, $|f''[A]^l|\leq n$.

Ramsey cardinals have $\kappa \rightarrow (\kappa)^{<\omega}_\omega$, which is equivalent to $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,1}$.

It seems to be similar to a "$n$-Rowbottom cardinal", but any definition I've seen of $\gamma$-Rowbottom cardinals are for $\gamma>\omega$, which makes me wonder if "$n$-Rowbottom" is just equivalent to Ramsey.

For example the definition from Kanamori's book:

"If $\omega < \gamma < \kappa$, then $\kappa $ is $\gamma$-Rowbottom iff $\kappa \rightarrow [\kappa]^{<\omega}_{\lambda,<\gamma}$ for any $\lambda<\kappa$"

I guess I'm asking if $n>1$ and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ implies $\kappa \rightarrow (\kappa)^{<\omega}_\omega$. I don't think it does, but it would be nice to see an example where cardinals with partition relations of this form are used.

Answer 1

Suppose (for example), that for every coloring $c:[\kappa]^{< \omega} \rightarrow \omega$, there is a set $H \in [\kappa]^\kappa$ such that $(\forall n \in \omega)|c([H]^n)| \leq 16$. Note that this implies that cofinality of $\kappa$ is at least $\omega_1$. Let us try to bring $16$ down to $1$. Assume that this is impossible for some coloring $c$. Then for every $\kappa$ sized subset $A \subseteq \kappa$, there is some $m$ and there are arbitrarily long sequences $s_1, s_2, \dots s_k \in [A]^{m}$ such that max$(s_i) <$ min$(s_{i+1})$ and $c(s_i) \neq c(s_{i+1})$. Define a new coloring $d:[\kappa]^{< \omega} \rightarrow \omega^{< \omega}$ such that for $s \in [\kappa]^{< \omega}$, $d(s)$ encodes the $c$-colors of all finite increasing sequences in $s$. Get an $H \in [\kappa]^{\kappa}$ such that, $(\forall n \in \omega)|d([H]^n)| \leq 16$. Let $s_1^{0}, s_1^{1}, s_2^{0}, s_2^1, \dots , s_5^0, s_5^1 \in [H]^m$ be a sequence of sets such that max of each set is less than the min of next one and $c(s_i^0) \neq c(s_i^1)$, for $1 \leq i \leq 5$. But this meanss $|d([H]^{5m}| \geq 32$.

Answer 2

This is not a complete answer, but clears up some of the confusion.

After a bit more research, the definition of $\kappa \rightarrow [\kappa]^{<\omega}_{\alpha,\beta}$ is not the one I stated. This is where the confusion over Rowbottom cardinals stemmed from.

The real definition is for every $f:[\kappa]^{<\omega}\rightarrow \alpha$ there is an $A\in [\kappa]^\kappa$ such that $|f"[A]^{<\omega}|\leq \beta$.

With this new definition, $f:[\kappa]^{<\omega}\rightarrow \omega$, $f(\eta)=\mbox{ length of } \eta$,

contradicts $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$, (as for any $A\in [\kappa]^\kappa$, $f"[A]^{<\omega}=\omega$). So that this never holds.

However, the question still stands, with the old definition in the last post, (which is a bit weaker) does this imply $\kappa$ is Ramsey? (this seems to be a slight generalization of $\kappa\rightarrow (\kappa)^{<\omega}_\omega$)