Worked solution: Independence of Suslin's Hypothesis from ZFC (Jensen, Solovay–Tennenbaum, 1971)

The diamond principle $\diamondsuit$ asserts the existence of a sequence $\langle A_\alpha : \alpha < \omega_1 \rangle$ with $A_\alpha \subseteq \alpha$ for each $\alpha$, such that for every $Z \subseteq \omega_1$, the set $\{ \alpha < \omega_1 : Z \cap \alpha = A_\alpha \}$ is stationary in $\omega_1$ (it meets every closed unbounded subset of $\omega_1$, so in particular it is uncountable and "large" in a precise topological sense, even though it can still have size $\aleph_1$ rather than all of $\omega_1$).

Jensen proved, through his fine-structural analysis of the constructible hierarchy $L$ described in the earlier steps of the continuum hypothesis proof, that $\diamondsuit$ holds whenever $V = L$: the same definability control that pins down cardinalities in $L$ also lets one build, level by level, a sequence of guesses that stays correct stationarily often.

$\diamondsuit$ is a strictly stronger hypothesis than $\mathrm{CH}$ (it implies $\mathrm{CH}$ but is not implied by $\mathrm{CH}$), and its real power is combinatorial rather than just about cardinal arithmetic: it supplies exactly the kind of "oracle" needed to guide a transfinite construction while avoiding every possible obstruction in advance, which the next step uses to build a Suslin tree.