 Solution

Write the Ehrenfeucht-Mostowski model as the Skolem hull of its order-indiscernible skeleton $(a_i{)}_{i\in \eta }$. Every element is $t(a_{i_1},\dots ,a_{i_r})$ for a Skolem term $t$. Choose supports for the elements of $X$ and let $J\subseteq \eta$ be their union. Then

When $\eta$ is well ordered, the type over $X$ of $t(a_{i_1},\dots ,a_{i_r})$ is determined by $t$ and the finite order pattern of the indices $i_k$ relative to $J$. There are at most $|L|$ terms and at most $|J|$ such finite patterns. Therefore the number of realized complete one-types is at most