Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 3 / iii

Write ${◊}_S^{\prime }$ for the countable-family assertion in the PDF; its prime is not a superscript $c$. A single stationary diamond at a regular cardinal sequence gives such families by taking singletons, so ${◊}_S\Rightarrow {◊}_S^{\prime }$.

Conversely enumerate each countable family as $E_{\alpha }(n)$, $n<\omega$, padding finite families and allowing the empty set as a default. Fix a bijection $\pi :\lambda \times \omega \to \lambda$. The previous part applied to $B(\xi )=\{\pi (\xi ,n):n<\omega \}$ gives a club set $D$ on which $\pi [\alpha \times \omega ]\subseteq \alpha$. Define, for each $n$, a single candidate sequenceSuppose no candidate sequence witnesses ${◊}_S$. For each $n$ choose $X_n\subseteq \lambda$ and a club set $C_n$ such that $D_{\alpha }^n\ne X_n\cap \alpha$ for every $\alpha \in S\cap C_n$. Code all the counterexamples intoThe countable-family hypothesis guesses $X$ on a stationary set. Choose a guessing $\alpha$ in the club set $D\cap {\bigcap }_n{C}_n$, and choose $n$ with $E_{\alpha }(n)=X\cap \alpha$. For every $\xi <\alpha$, closure under $\pi$ givesThus $D_{\alpha }^n=X_n\cap \alpha$, contradicting $\alpha \in C_n$. At least one candidate is a diamond sequence, provingThis is the countable-family diamond equivalence, for every regular uncountable $\lambda$ and stationary $S\subseteq \lambda$.