Ellentuck Borel sets are completely Ramsey

Every Borel set for the Ellentuck topology is completely Ramsey. First, an acceptance-and-rejection fusion makes every open set homogeneous after thinning the infinite reservoir while keeping its finite stem. Every nowhere-dense set is consequently completely Ramsey-null, and a fusion handling all subsets of each finite selected prefix shows that countable unions of such null sets remain null. Finally, sets differing from an open set by a meagre set form a sigma-algebra containing the open sets. Thus every Borel set has the required stem-preserving homogeneous refinement.

In the Ellentuck topology, suppose every open set is completely Ramsey and every meagre set is completely Ramsey-null. If $E△U$ 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.