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Embedding Borel graphs into grids of asymptotically optimal dimension | Synapse

March 15, 2026

Embedding Borel graphs into grids of asymptotically optimal dimension

Key Points

To investigate the embedding of Borel graphs into asymptotically optimal dimensional grids.
Analyzed finite subgraphs of Borel graphs
Developed a Borel embedding into Schreier graphs
Examined Borel actions related to polynomial growth rates.
Established that Borel graphs embed into dimensional grids with diagonals.
Strengthened prior results on embedding complexity by introducing polynomial growth rates.

Abstract

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.

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Cite This Study

Bernshteyn et al. (Fri,) studied this question.

synapsesocial.com/papers/69b5ff6e83145bc643d1bf28 https://doi.org/https://doi.org/10.1090/proc/17597

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