Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 6 / c

For each $\xi <\kappa$, choose a maximal antichain in a forcing order below $p$ deciding a witness $\eta >\xi$ in $\dot{C}$. Write the decided values as ${\eta }_{\xi ,q}$, one for each condition $q$ in that antichain. The $\kappa$-chain condition for forcing gives fewer than $\kappa$ possible values. Since $\kappa$ is regular in $M$, chooseThe possible-values lemma for chain-condition forcing ensuresIn $M$ take the club set of limit closure pointsIt is unbounded by repeatedly closing any starting ordinal under $g$ for countably many steps, and closed by taking increasing limits. For $\delta \in D$, the forced displayed property gives points of $\dot{C}$ arbitrarily high below $\delta$. Its forced closedness yields $\delta \in \dot{C}$. ThereforeThis proves the ground-model club containment lemma. The name on the right is $\dot{C}$, as printed in the PDF.

Fix $p$ and a name $\dot{f}$ such that $p$ forces a two-coloring of $[\lambda {]}^{<\omega }$. For each ground-model finite set $a\subseteq \lambda$, record its full forcing decision patternThere are at most $3^{|\mathbb{P}|}<\lambda$ such patterns, since $\lambda$ is a strong limit cardinal and $|\mathbb{P}|<\lambda$. For finite $\mathbb{P}$ the number of patterns is finite, also below $\lambda$. Encode the patterns by an ordinal $\beta <\lambda$ and apply the assumed finite-subset partition property to obtain $H\subseteq \lambda$ of size $\lambda$ such that $F$ is constant on $[H{]}^n$ for every $n<\omega$.

For each $n$, choose one $a_n\in [H{]}^n$. Conditions below $p$ deciding $\dot{f}(a_n)$ are dense. Whenever one such condition decides the value, equality of patterns means it forces that same value at every $a\in [H{]}^n$. A generic filter containing $p$ meets this dense set for each $n$, so $f$ is constant on each $[H{]}^n$ in the extension. The constants may differ with $n$, exactly as required.

The forcing has size below the regular $\lambda$, hence satisfies the $\lambda$-chain condition for forcing and preserves its cardinality. The ground-model set $H$ consequently still has size $\lambda$. Since the argument works for every name and condition,This is small forcing preservation of finite-subset partition properties; no closure assumption on the forcing is required.