Universal geometric non-embedding of random regular graphs

Abstract

Let $\Delta \ge 3$ be fixed, $n\ge n_{\Delta }$ be a large integer. It is a classical result that $\Delta$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c\log n$, for some universal constant $c$. We show that for typical $\Delta$-regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random $\Delta$-regular graph $G$ on $n$ vertices, it holds with high probability: there is no normed space of dimension less than $c\log n$ which admits a geometric graph isomorphic to $G$. The proof is based on a seeded multiscale $\epsilon$--net argument.

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