Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 4 / i

The partition relation $\kappa \to (\lambda {)}_{\beta }^{\alpha }$ means that for every coloring $c:[\kappa {]}^{\alpha }\to \beta$, there is $H\subseteq \kappa$ with $|H|=\lambda$ such that $c$ is constant on $[H{]}^{\alpha }$. Here $[X{]}^{\alpha }$ denotes the subsets of $X$ of cardinality $\alpha$, and the subscript is the number of colors. The domain consists of subsets of size $\alpha$, not ordered tuples, and the definition also makes sense for infinite arity.

The stationary-indexed club principle ${♣}_S$ asserts a sequence $⟨A_{\delta }:\delta \in S\cap \mathrm{Lim}({\omega }_1)⟩$ with each $A_{\delta }\subset \delta$ cofinal of order type $\omega$, such that for every uncountable $X\subseteq {\omega }_1$ some $\delta \in S$ satisfies $A_{\delta }\subseteq X$. It predicts a cofinal ladder contained in the target, rather than predicting the entire initial segment of the target as diamond does. For a stationary $S$ with successor members, only its stationary limit-ordinal part is relevant; countable limit ordinals have cofinality $\omega$. This is the restricted version of the club principle.

A tree with unique limits has no distinct nodes at a limit level with the same history. Precisely, for a limit ordinal $\delta$ and $s,t\in T_{\delta }$,Here $s\upharpoonright \beta$ denotes the unique predecessor of $s$ of height $\beta$. This is uniqueness of a limit node when it exists, not a requirement that every cofinal chain in a partial order below a limit level acquire a limit node.