= Solution
In the usual tail-normalized version, an <Ulam matrix on omega-one> is a family $A_{\alpha,n}$ such that, for each $\alpha<\omega_1$, the <sets> indexed by $n<\omega$ partition the tail $(\alpha,\omega_1)$, while for each fixed $n$ the <sets> indexed by $\alpha<\omega_1$ are pairwise disjoint. Empty cells are allowed; harmless bounded-tail variants give the same applications.
For a concrete realization, choose <injections> $e_\beta:\beta\to\omega$ for every <countable ordinal> $\beta$, and <set>
$$
\boxed{A_{\alpha,n}=\{\beta>\alpha:e_\beta(\alpha)=n\}.}
$$
For fixed $\alpha$ each $\beta>\alpha$ chooses exactly one $n$; for fixed $n$, injectivity of $e_\beta$ prevents one $\beta$ from belonging to two cells. This verifies the two matrix conditions explicitly.
Back to article page
