  In this paper we study the combinatorics of free Borel actions of the group
$\mathbb Z^d$ on Polish spaces. Building upon recent work by Chandgotia and
Meyerovitch, we introduce property $F$ on $\mathbb Z^d$-shift spaces $X$ under
which there is an equivariant map from any free Borel action to the free part
of $X$. Under further entropic assumptions, we prove that any subshift $Y$
(modulo the periodic points) can be Borel embedded into $X$. Several examples
satisfy property $F$ including, but not limited to, the space of proper
$3$-colourings, tilings by rectangles (under a natural arithmetic condition),
proper $2d$-edge colourings of $\mathbb Z^d$ and the space of bi-infinite
Hamiltonian paths. This answers questions raised by Seward, and Gao-Jackson,
and recovers a result by Weilacher and some results announced by
Gao-Jackson-Krohne-Seward.
