Metric embeddings of cubes into dense subsets of cubes

Abstract: Fix $k \in \mathbb{N}$ and $0 < δ< 1$. We study how large $N$ must be so that every $δ$-dense subset $\mathcal{D} \subset {0,1}^N$ (meaning $|\mathcal{D}| \geq δ2^N$) contains the image of a metric embedding $f: {0,1}^k \to \mathcal{D}$. We study three variants. For a $(1+\varepsilon)$-bi-Lipschitz map $f$ with fixed $\varepsilon > 0$, we show $N = O(\varepsilon^{-2} \log(1/δ) k^3)$. For an isometric map with arbitrary rescaling (undistorted), we show $N = \log(1/δ) e^{Ω(k)}$ and conjecture $N = \log(1/δ) e^{O(k)}$. For an isometric map with bounded rescaling we show $N = \exp[\log(1/δ) e^{Θ(k)}]$. As a geometric application, we obtain a nonpositive Alexandrov curvature counterpart to the work of Bartal-Linial-Mendel-Naor on the nonlinear Dvoretzky problem. It is known that any subset of ${0,1}^N$ embedding with bi-Lipschitz distortion $< α$ into a metric space of nonnegative Alexandrov curvature must satisfy $|\mathcal{D}| \lesssim 2^{N(1-Ω(α^{-2}))}$. Work of Gromov and Kondo shows that this approach does not extend to CAT(0) targets. We prove that for every $N \gtrsim α^6 \geq 1$, any $\mathcal{D} \subset {0,1}^N$ embedding with distortion $< α$ into a CAT(0) space must satisfy $|\mathcal{D}| \lesssim 2^{N(1-Ω(α^{-4}))}$, via a completely different approach. Similar results hold for targets of nontrivial Enflo type. Finally, we prove the density analogue of a coloring theorem of Rodl-Sales: we give bounds for $(1+\varepsilon)$-bi-Lipschitz embeddings of the path ${1,\ldots,k}$ into dense subsets of ${1,\ldots,N}$ (improving a bound of Dumitrescu), and prove similar bounds for binary tree metrics.