the k-SRP stationary Ramsey property

Question

I would like to know why here we have $$\kappa<\omega$$

in

"where we partition all of $\kappa<\omega$"

Answer 1

First of all $\kappa<\omega$ is a typo and should be $[\kappa]^{<\omega}$.

Let me introduce some notation: $$\kappa\to(\alpha)^\gamma_\delta$$ means that every function $f\colon[\kappa]^\gamma\to\delta$ has a homogeneous set of order type $\alpha$, or in other words that there exist $H\in[\kappa]^\alpha$ such that $f\upharpoonright H^\gamma$ is constant. In plain English this means that whenever we colour the subsets of $\kappa$ of cardinality $\gamma$ using $\delta$-many colours, there is an $H\subseteq\kappa$ of order type $\alpha$, such that all all subsets of $H$ of cardinality $\gamma$ have the same colour.

Similarly $$\kappa\to(\alpha)^{<\omega}_\delta$$ means that for every $f\colon[\kappa]^{<\omega}\to\delta$ there is $H\in[\kappa]^\alpha$ such that $f\upharpoonright H^n$ is constant for every $n$.

With this notation, based on the description between brackets, what the author is calling $\kappa\to\omega$ is $\kappa\to(\omega)^{<\omega}_2$ (such cardinals are known as Ramsey cardinals).

To answer the question raised in the comments being $k$-SRP now means that $\kappa\to(\kappa)^k_2$ and furthermore the homogenous $H$ can be chosen to be stationary in $\kappa$. In particular $\kappa$ is $2$-SRP if $\kappa\to(\kappa)^2_2$ and for every $f\colon[\kappa]^2\to 2$ the homogeneous set for $f$ can be chosen to be stationary in $\kappa$. Since $\kappa\to(\kappa)^2_2$ is equivalent to $\kappa$ being weakly compact (see Theorem 7.8 in Kanamori's book) we immediately have that the least $2$-SRP cardinal is at least as big as the least weakly compact one.