  We prove that for any homogeneous structure $\mathbf{K}$ in a language with
finitely many relation symbols of arity at most two satisfying SDAP$^+$ (or
LSDAP$^+$), there are spaces of subcopies of $\mathbf{K}$, forming subspaces of
the Baire space, in which all Borel sets are Ramsey. Structures satisfying
SDAP$^+$ include the rationals, the Rado graph and more generally, unrestricted
structures, and generic $k$-partite graphs, the latter three types with or
without an additional dense linear order. As a corollary of the main theorem,
we obtain an analogue of the Nash-Williams Theorem which recovers exact big
Ramsey degrees for these structures, answering a question raised by Todorcevic
at the 2019 Luminy Workshop on Set Theory. Moreover, for the rationals and
similar homogeneous structures our methods produce topological Ramsey spaces,
thus satisfying analogues of the Ellentuck theorem.
