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Embedding Borel graphs into grids of asymptotically optimal dimension

Anton Bernshteyn, Jing Yu

TL;DR

The paper advances the study of Borel embeddings for graphs of polynomial growth by proving that the Borel embedding dimension emb_B(G) is at most a constant multiple of the ordinary embedding dimension emb(G). It introduces a key lemma producing $R$-locally injective $1$-Lipschitz maps into $\mathbb{Z}^{4d}$, leveraging the Borel asymptotic dimension and asymptotic separation index, and then combines this with a global coarse embedding into a Schreier graph of a free $\mathbb{Z}^n$-action to achieve distance-preserving Borel embeddings into grids with diagonals. This yields a uniform, distance-respecting embedding framework for Borel graphs, with immediate corollaries for classes defined by induced cycles and forbidden minors. The results illuminate the structure of Borel graph embeddings and connect descriptive set-theoretic methods with large-scale geometric techniques.

Abstract

Let $G$ be a Borel graph all of whose finite subgraphs embed into the $d$-dimensional grid with diagonals. We show that then $G$ itself admits a Borel embedding into the Schreier graph of a free Borel action of $\mathbb Z^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$ is replaced by $O(ρ\log ρ)$, where $ρ$ is the polynomial growth rate of $G$.

Embedding Borel graphs into grids of asymptotically optimal dimension

TL;DR

The paper advances the study of Borel embeddings for graphs of polynomial growth by proving that the Borel embedding dimension emb_B(G) is at most a constant multiple of the ordinary embedding dimension emb(G). It introduces a key lemma producing -locally injective -Lipschitz maps into , leveraging the Borel asymptotic dimension and asymptotic separation index, and then combines this with a global coarse embedding into a Schreier graph of a free -action to achieve distance-preserving Borel embeddings into grids with diagonals. This yields a uniform, distance-respecting embedding framework for Borel graphs, with immediate corollaries for classes defined by induced cycles and forbidden minors. The results illuminate the structure of Borel graph embeddings and connect descriptive set-theoretic methods with large-scale geometric techniques.

Abstract

Let be a Borel graph all of whose finite subgraphs embed into the -dimensional grid with diagonals. We show that then itself admits a Borel embedding into the Schreier graph of a free Borel action of . This strengthens an earlier result of the authors, in which is replaced by , where is the polynomial growth rate of .
Paper Structure (6 sections, 18 theorems, 18 equations, 1 figure)

This paper contains 6 sections, 18 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theo:coarse']} assuming Lemma \ref{['lemma:key']}
  3. Proof of Lemma \ref{['lemma:key']}
  4. Borel asymptotic dimension and asymptotic separation index
  5. Constructing locally injective Lipschitz functions
  6. Further remarks

Key Result

Theorem 1.1

If $F$ is a finite graph of growth rate $\rho$, then $F$ has an embedding into $\mathbb{Z}^{d}_\infty$ for some $d = O(\rho \log \rho)$.

Figures (1)

  • Figure 1: A fragment of $\mathbb{Z}^{2}_\infty$.

Theorems & Definitions (29)

  • Theorem 1.1: Krauthgamer--Lee krauthgamer2007intrinsic
  • Definition 1.2: Embedding dimension
  • Theorem 1.3: Krauthgamer--Lee krauthgamer2007intrinsic
  • Example 1.4: Schreier graphs
  • Definition 1.5: Borel graphs
  • Definition 1.6: Borel embedding dimension
  • Theorem 1.7: AB--JY BernshteynYu
  • Theorem 1.8: Main result: Borel embedding dimension
  • Corollary 1.9: Induced cycles and forbidden minors
  • Theorem 1.10: Embeddings with low distance reduction
  • ...and 19 more