# Ramsey-theoretic properties of Erdős cardinals

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

• Since the quoted claim says "an ω-Erdos cardinal" and not "the ω-Erdos cardinal", in this question's context is this referring to η being a ω-club-Erdos cardinal?
– C7X
Jun 29, 2022 at 20:21
• @C7X Yes, the paper uses the definition that I call ω-club-Erdos. Jun 29, 2022 at 20:51
• @bof Corrected. Jun 30, 2022 at 10:45