Greatly Erdős and Erdős cardinals

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?