Sharpe and Welch 2011 define $\alpha$-weakly Erdős and greatly Erdős cardinals as follows:

Let $\kappa$ have uncountable cofinality, and let $\mathcal{A}$ be a $\kappa$-structure, $X \subseteq \kappa$. Suppose that we have $t_{\mathcal{A}} (X) = \{\alpha \in \kappa \cap lim : \text{there exists a set $I \subseteq \alpha \cap X$ of good indiscernibles for $\mathcal{A}$ cofinal in $\alpha$}\}$.

Let $\kappa$ be a cardinal of uncountable cofinality. Set $F_0 = C_\kappa$, the club filter for $\kappa$, $F_{\alpha + 1} = \{X \subseteq \kappa : \text{$X \supseteq t_{\mathcal{A}} (Y)$, for some $Y \in F_\alpha$ and $\kappa$-structure $\mathcal{A}$}\}$, $F_\alpha = \{X \subseteq \kappa : \text{$X \supseteq \Delta Y$ for some $Y \subseteq \bigcup_{\beta \lt \alpha} F_\beta$, $|Y| \le \kappa$} \}$ for the limit $\alpha \lt \kappa^+$. $\kappa$ is $\alpha$-weakly Erdős iff $\emptyset \notin F_\alpha$.

Equivalently, $F_{\alpha + 1} = \{X \subseteq \kappa : \text{$X \supseteq t_f (Y)$, for some $Y \in F_\alpha$ and regressive function $f: [\kappa]^{\lt \omega} \to \kappa$}\}$, where $t_f (X) = \{\alpha \in \kappa \cap lim : \text{$\alpha$ is the limit of an $f$-homogeneous set $I \subseteq \alpha \cap X$}\}$ (regressive means that $f(a) \lt min(a)$ for every $a \in dom(f)$). A cardinal $\kappa$ is greatly Erdős if and only if it is $\alpha$-weakly Erdős for every $\alpha \lt \kappa^+$. Below I will say $\alpha$-greatly Erdős instead of $\alpha$-weakly Erdős.

A cardinal $\kappa$ is $\beta$-Erdős if, for every club $C \subseteq \kappa$ and every regressive function $f: [\kappa]^{\lt \omega} \to \kappa$ there is an $f$-homogeneous set of order type $\beta$.

If $\kappa$ is 2-greatly Erdős, is it $\beta$-Erdős for every $\beta \lt \kappa$? I don't understand the proof of lemma 4.33 of the paper linked above. Can it be translated to use Ramsey-theoretic characterizations instead of the model-theoretic ones used in the paper?



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