Year: 2026
Type: article
Abstract: A graph $$G=(V,E)$$ G = ( V , E ) is geometrically embeddable into a normed space X when there is a mapping $$\zeta :V\rightarrow X$$ ζ : V → X such that $$\Vert \zeta (v)-\zeta (w)\Vert _X\leqslant 1$$ ‖ ζ ( v ) - ζ ( w ) ‖ X ⩽ 1 if and only if $$\{v,w\}\in E$$ { v , w } ∈ E , for all distinct $$v,w\in V$$ v , w ∈ V . Our result is the following universal threshold for the embeddability of trees. Let $$\Delta \geqslant 3$$ Δ ⩾ 3 , and let N be sufficiently large in terms of $$\Delta $$ Δ . Every N –vertex tree of maximal degree at most $$\Delta $$ Δ is embeddable into any normed space of dimension at least $$64\,\frac{\log N}{\log \log N}$$ 64 log N log log N , and complete trees are non-embeddable into any normed space of dimension less than $$\frac{1}{2}\,\frac{\log N}{\log \log N}$$ 1 2 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.
Source: COMBINATORICA
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W7161940455