Metric embeddings of cubes into dense subsets of cubes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910041613795328 |
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| author | Karamanlis, Miltiadis Kravaris, Cosmas |
| author_facet | Karamanlis, Miltiadis Kravaris, Cosmas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04644 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Metric embeddings of cubes into dense subsets of cubes Karamanlis, Miltiadis Kravaris, Cosmas Combinatorics Metric Geometry 05D10, 46B85 (Primary), 53C23, 05C12, 46B20 (Secondary) 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. |
| title | Metric embeddings of cubes into dense subsets of cubes |
| topic | Combinatorics Metric Geometry 05D10, 46B85 (Primary), 53C23, 05C12, 46B20 (Secondary) |
| url | https://arxiv.org/abs/2603.04644 |