= Solution
Choose $\aleph_2$ distinct <cofinal branches> of the <Kurepa tree> and regard them as subsets of its underlying set $\omega_1$. Let $\mathcal F$ be this family. Fix an ordinal $\alpha<\omega_1$. The countably many node codes below $\alpha$ have heights bounded by some $\delta<\omega_1$. A <cofinal branch> meets level $\delta$, and its node there determines all its predecessors, hence all its nodes with codes below $\alpha$. Since level $\delta$ is countable,
$$
\boxed{|\mathcal F|=\aleph_2,\qquad
|\{X\cap\alpha:X\in\mathcal F\}|\leq\aleph_0\quad(\alpha<\omega_1).}
$$
This is the <Kurepa-family hypothesis>. The given compatibility between the ordinal codes and the tree order ensures in particular that the branches are consistently viewed as subsets of $\omega_1$; boundedness of the heights of the countably many codes is what the argument uses.
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