= Solution
We prove the <Ehrenfeucht-Mostowski theorem> by <ultraproducts>, with no application of <Ramsey theorem>. Expand the infinite structure $M$ by <Skolem functions>, choose distinct elements $a_0,a_1,\ldots$, and choose a <nonprincipal ultrafilter> $D$ on $\omega$. Let $I$ be the desired total index order and put $J=\omega^I$.
For every finite ordered <subset> $F=\{i_1<\cdots<i_n\}$, define an <ultrafilter> $D_F$ on $\omega^F$ by the ordered <Fubini product of ultrafilters>: a <set> $B$ belongs to $D_F$ exactly when
$$
(Dm_1)(Dm_2)\cdots(Dm_n)\quad (m_1,\ldots,m_n)\in B.
$$
Here $(Dm)\psi(m)$ means $\{m:\psi(m)\}\in D$, with the quantifiers nested in the displayed order. The resulting collection is an <ultrafilter>, by induction using the <ultrafilter> laws for negation and conjunction. For $F\subseteq H$, the inverse image of $B\subseteq\omega^F$ under coordinate projection belongs to $D_H$ exactly when $B\in D_F$: quantifiers on unused coordinates leave a truth value unchanged.
Consequently there is a well-defined <ultrafilter> on the <Boolean algebra> of finite-coordinate cylinders in $J$, declaring a cylinder large by this test. Coherence proves finite <intersection> closure and ensures that the empty cylinder is not large. Extend its generated filter to an <ultrafilter> $U$ on all <subsets> of $J$. Form the <ultrapower> $N=(M^*)^J/U$, and for $i\in I$ let $b_i$ be the class of $s\mapsto a_{s(i)}$.
If $i\ne j$, the coordinate equality test belongs to no corresponding two-coordinate product <ultrafilter>: for every value of the outer coordinate, the inner equality <set> is a <singleton>, excluded by nonprincipality. Hence the $b_i$ are distinct. For any <first-order formula> $\varphi(x_1,\ldots,x_n)$ and any $i_1<\cdots<i_n$, the <Łoś theorem> gives
$$
N\models\varphi(b_{i_1},\ldots,b_{i_n})
\quad\Longleftrightarrow\quad
(Dm_1)\cdots(Dm_n)\ M^*\models\varphi(a_{m_1},\ldots,a_{m_n}).
$$
The right side is independent of the actual increasing index tuple, so the $b_i$ form an <order-indiscernible sequence>. Constant <functions> embed $M^*$ elementarily into $N$. Take the <Skolem hull> of the generators. The <Tarski-Vaught test> gives an <elementary substructure>, and transporting terms along an index-order automorphism gives a well-defined automorphism of the hull, with inverse obtained by transporting along the inverse index automorphism. Equality of term representations is preserved by indiscernibility. \b[This is the indiscernible and automorphism conclusion of Ehrenfeucht–Mostowski, obtained entirely from <ultrafilters> and Łoś's theorem.] The ordered Fubini products need not be symmetric; order indiscernibility is exactly what their coherence supplies.
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