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Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.

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Ramsey-theoretic properties of Erdős cardinals

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 $\kappa$-li …
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