 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 thatThe 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.