Embedding Borel graphs into grids of asymptotically optimal dimension

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 ) ZO(d) . This strengthens an earlier result of the authors, in which O ( d ) O(d) is replaced by O ( ρ log ⁡ ρ ) O(ρ log ρ ) , where ρ ρ is the polynomial growth rate of G G .