= Solution
A precise form of the <generalized delta-system lemma> is this. Let $\mu<\theta$ be infinite <cardinals>, with $\theta$ regular and uncountable, and suppose $|[\alpha]^{<\mu}|<\theta$ for every $\alpha<\theta$. Every family $\mathcal A$ of $\theta$ distinct sets, each of cardinality less than $\mu$, contains a subfamily $\mathcal B$ of size $\theta$ and a set $r$ such that
$$
\boxed{x\ne y\in\mathcal B\ \Longrightarrow\ x\cap y=r.}
$$
The members form a <delta-system> with root $r$. A sufficient usual arithmetic hypothesis is $\alpha^\mu<\theta$ for all infinite <cardinals> $\alpha<\theta$. The finite-set case $\mu=\omega$ needs only regular uncountable $\theta$, since finite <subsets> of each $\alpha<\theta$ form a family of size less than $\theta$. This is the form used for the finite-support collapse below.
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