Metric embeddings of cubes into dense subsets of cubes
Fix $k \in \mathbb{N}, 0 < \delta < 1$. We study how large $N$ must be so that every $\delta$-dense subset $\mathcal{D} \subset \{0,1\}^N$ (meaning $|\mathcal{D}|\geq \delta 2^N$) contains the image of a metric embedding $f: \{0,1\}^k \to \mathcal{D}$. We study $3$ variants: For a $(1+\varepsilon)$-bi-Lipschitz map $f$ for a fixed $\varepsilon>0$, we show that $N = O(\varepsilon^{-2}\log(1/\delta) k^3)$. For an isometric map $f$ with arbitrary rescaling (i.e. undistorted), we show that $N = \log(1/\delta) e^{\Omega(k)}$. For an isometric map $f$ with bounded rescaling we show that $N = \exp{[\log(1/\delta)e^{\Theta(k)}]}$. Regarding the path space, we prove the density analog of a coloring theorem of R\"odl--Sales. We give bounds for $(1+\varepsilon)$-bi-Lipschitz embeddings of the path $[k] = \{1,...,k\}$ into dense subsets of the path $[N] = \{1,...,N\}$, improving a bound of Dumitrescu. We prove similar bounds for the binary tree space, using the tree replicas theorem of Pach--Solymosi--Tardos. As a geometric application we obtain a non-positive Alexandrov curvature counterpart to the work of Bartal--Linial--Mendel--Naor on the nonlinear Dvoretzky problem who showed that any $\mathcal{D} \subset \{0,1\}^N$ that embeds with bi-Lipschitz distortion $<\alpha$ into a metric space of non-negative Alexandrov curvature must be small, namely, necessarily $|\mathcal{D}| \lesssim 2^{N(1-\Omega(\alpha^{-2}))}$. We prove that for every $N\gtrsim \alpha^{6}\ge 1$, any $\mathcal{D} \subset \{0,1\}^N$ that embeds with distortion $<\alpha$ into some metric space of non-positive Alexandrov curvature must satisfy $|\mathcal{D}| \lesssim 2^{N(1-\Omega(\alpha^{-4}))}$ via an approach which is entirely different from that of Bartal--Linial--Mendel--Naor. We also show that nontrivial metric type and non-universality are preserved by taking finite unions of subspaces.