A universal threshold for geometric embeddings of trees

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.