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A universal threshold for geometric embeddings of trees

Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros

TL;DR

The paper establishes a universal threshold for geometrically embedding trees into finite-dimensional normed spaces. It introduces a randomized embedding that attaches independent unit-ball vectors to edges and adds Gaussian regularization, then analyzes the embedding using small-ball and concentration tools derived from the slicing Bourgain framework. A key contribution is showing that for any bounded-degree tree on $N$ vertices, embeddability into a space of dimension $\Theta\big(\frac{\log N}{\log\log N}\big)$ is both possible (upper threshold) and sometimes impossible (lower threshold) up to universal constants, with precise bounds $\alpha_0$ and $\alpha_1$ satisfying $\tfrac{1}{2} \le \alpha_0 < \alpha_1 \le 64$. The methods sophisticatedly combine isotropic log-concave geometry, thin-shell and anticoncentration results, and an asymmetric Lovász local lemma to control dependencies, yielding a constructive proof that complete trees achieve the threshold and that the bound is tight within constants. The work has implications for intersection graphs of translates of convex bodies and advances understanding of dimensional trade-offs in geometric graph representations.

Abstract

A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $ζ: V\to X$ such that $\|ζ(v)-ζ(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the following universal threshold for the embeddability of trees. Let $Δ\geqslant 3$, and let $N$ be sufficiently large in terms of $Δ$. Every $N$--vertex tree of maximal degree at most $Δ$ is embeddable into any normed space of dimension at least $64\,\frac{\log N}{\log\log N}$, and complete trees are non-embeddable into any normed space of dimension less than $\frac{1}{2}\,\frac{\log N}{\log\log N}$. In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.

A universal threshold for geometric embeddings of trees

TL;DR

The paper establishes a universal threshold for geometrically embedding trees into finite-dimensional normed spaces. It introduces a randomized embedding that attaches independent unit-ball vectors to edges and adds Gaussian regularization, then analyzes the embedding using small-ball and concentration tools derived from the slicing Bourgain framework. A key contribution is showing that for any bounded-degree tree on vertices, embeddability into a space of dimension is both possible (upper threshold) and sometimes impossible (lower threshold) up to universal constants, with precise bounds and satisfying . The methods sophisticatedly combine isotropic log-concave geometry, thin-shell and anticoncentration results, and an asymmetric Lovász local lemma to control dependencies, yielding a constructive proof that complete trees achieve the threshold and that the bound is tight within constants. The work has implications for intersection graphs of translates of convex bodies and advances understanding of dimensional trade-offs in geometric graph representations.

Abstract

A graph is geometrically embeddable into a normed space when there is a mapping such that if and only if , for all distinct . Our result is the following universal threshold for the embeddability of trees. Let , and let be sufficiently large in terms of . Every --vertex tree of maximal degree at most is embeddable into any normed space of dimension at least , and complete trees are non-embeddable into any normed space of dimension less than . In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.

Paper Structure

This paper contains 15 sections, 8 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. Notation
  4. Isotropic random vectors, and normed spaces in isotropic position
  5. Asymptotic geometric analysis
  6. Gaussian concentration
  7. John's theorem
  8. Lovász local lemma
  9. Preparatory lemmas
  10. Proof of Theorem \ref{['thm:complete-tree-embedding']}
  11. Construction of the random embedding
  12. Auxiliary events
  13. The "path" graph
  14. Application of asymmetric Lovász local lemma
  15. Completion of the proof of part $(B)$ of Theorem \ref{['thm:complete-tree-embedding']}

Key Result

Theorem 1.1

There exist absolute positive constants $\alpha_0<\alpha_1$, and for every integer $\Delta\geqslant 3$ a positive integer $N_0=N_0(\Delta)$ depending only on $\Delta$ with the following property. Let $X$ be a finite-dimensional normed space, and let $N\geqslant N_0$ be an integer.

Theorems & Definitions (21)

  • Theorem 1.1: Universal threshold for embedding of trees
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2: Thin-shell concentration for log-concave measures
  • Theorem 2.3: Anticoncentration for log-concave measures via slicing
  • Remark 2.4
  • proof
  • Lemma 2.7: Asymmetric Lovász local lemma
  • ...and 11 more