= Baire-property reduction for completely Ramsey families
{c}
In the <Ellentuck topology>, suppose every <open set> is <completely Ramsey> and every <meagre set> is <completely Ramsey-null>. If $E\triangle U$ is contained in a meagre set for some open $U$, first refine a basic <neighbourhood> without changing its stem to avoid that exceptional set, and then refine again to be homogeneous for $U$. The resulting neighbourhood is homogeneous for $E$. Sets with such an open representative form a <sigma-algebra>: countable unions collect the meagre errors, and complements replace $U^c$ by its interior, differing only on the nowhere-dense boundary of $U$. This proves that all star-<Borel sets> are completely Ramsey.
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