Let G G be a Borel graph all of whose finite subgraphs embed into the d d /dimensional grid with diagonals. We show that then G G itself admits a Borel embedding into the Schreier graph of a free Borel action of Z O (d) Z^O (d). This strengthens an earlier result of the authors, in which O (d) O (d) is replaced by O (ρ log ρ) O (), where ρ is the polynomial growth rate of G G.
Bernshteyn et al. (Fri,) studied this question.