Ramsey-theoretic properties of Erdős cardinals

Question

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 $\alpha \lt \eta \le \zeta$?

Answer 1

This answer will answer question 1 and almost answer question 2, giving a positive answer to question 2 when $\beta$ is a limit ordinal.

The linked paper cites theorem 6.1 of Schmerl's "On $\kappa$-like structures which embed stationary and closed unbounded subsets" (Annals of Mathematical Logic vol. 10, iss. 3--4, 1976) for the proof that the least $\beta$ Erdős cardinal is $\beta$-club Erdős. (This paper uses $\alpha$ for the order type of the subset, but to avoid a variable clash I will use $\beta$ for the order type here.)

There seem to be only two properties of $\kappa(\beta)$ used in the proof, which are choosing $h_\nu:[\nu]^{<\omega}\to 2$ for all $\nu<\kappa(\beta)$, and the choice of a set $X$ of indiscernibles of order type $\beta$. In particular the second one can be dealt with using theorem 5 of Drake's "A fine hierarchy of partition cardinals" (Fundamenta Mathematicae vol. 81, iss. 3, 1974), which states that if for some limit ordinal $\beta$ and some $\alpha\geq 2$, a cardinal $\eta$ is least such that $\eta\rightarrow(\beta)^{<\omega}_\alpha$, then a familiar indiscernibility property like for $\beta$-Erdős cardinals holds: any structure of length $<\eta$ which has a subset well-ordered with order type $\eta$ has a set of indiscernibles of order type $\beta$. (For a limit ordinal $\beta$, Drake calls a cardinal $\eta$ a $\beta$-partition cardinal if there exists some $\alpha\geq 2$ such that $\eta$ is least where $\eta\rightarrow(\beta)^{<\omega}_\alpha$.)

So the proof can be modified as follows: at the beginning of the proof choose an arbitrary cardinal $\alpha\geq 2$ and limit ordinal $\beta\geq\omega$; replace all instances of $\kappa(\beta)$ with $\eta$, the least cardinal such that $\eta\rightarrow(\beta)^{<\omega}_\alpha$; replace $h_\nu:[\nu]^{<\omega}\to 2$ with $h_\nu:[\nu]^2\to\alpha$; and use Drake's theorem to get the set $X$ of indiscernibles of order type $\beta$. Most of the proof proceeds by using indiscernibility of various sequences (for example $\{\max(b_\nu\cap C)\mid\nu<\beta\}$ in the second paragraph and $\{d_\nu\mid\nu<\beta\} = \{f_m(b_{(m+1)\nu},\ldots,b_{(m+1)\nu+m})\mid\nu<\beta\}$ for $h_{c_0}$-homogeneity at the end), and nowhere in the proof seems to depend on the codomain of $h_\nu$ being $2$.