The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous set of order type $\beta$. Since that definition only works for one ordinal $\eta$ for each $\beta$, it seems desirable to generalize that definition. Another definition is (equivalent to) the following: $\eta$ is $\beta$-Erdős if, for every club $C \subseteq \eta$ and every regressive function $f: [\eta]^{\lt \omega} \to \eta$ (regressive means that $f(a) \lt min(a)$ for every $a \in dom(f)$) there is an $f$-homogeneous set of order type $\beta$. Let's call the latter definition $\beta$-club Erdős. I can prove that $\zeta \to (\beta)^{\lt \beta}_2$ for any $\zeta$ greater than or equal to the $\beta$-Erdős cardinal for every $\beta$. Conversely, according to this paper, the least $\omega$-club Erdős is the $\omega$-Erdős cardinal. This follows from the following stronger claim in that paper:

if $\alpha \ge 2$ is a cardinal and there is a cardinal $\eta$ such that $\eta \to (\omega)^{\lt \omega}_\alpha$, then the least such cardinal $\eta$ is an $\omega$-Erdős cardinal (and is greater than $\alpha$.)

Question 1: **What is the proof of the quoted claim?**

Question 2: **Does this generalize to $\beta \gt \omega$?** That is, if $\zeta \to (\beta)^{\lt \omega}_\alpha$, must there be a $\beta$-club Erdős cardinal $\eta$ such that $\alpha \lt \eta \le \zeta$?