every edge in ${\{0,1\}^{n}}$ are mapped to close vertices in ${\mathcal{B}_{n}}$. The key building block we use is a classical partition of the vertices of ${\{0,1\}^{n}}$ to symmetric chains, due to De Bruijn, Tengbergen, and Kruyswijk [@BTK51], where a symmetric chain is a path $\{c_k, c_{k+1}, \ldots, c_{n-k}\}$ in ${\{0,1\}^{n}}$, such that each $c_i$ has Hamming weight $i$ (see Figure \[fig:the partition\]).

As a first step, we study the chains in the partition of De Bruijn [et al.]{}  Roughly speaking, we show[^7] that adjacent chains move closely to each other. More precisely, if two adjacent vertices $x$ and $y$ belong to different chains, then $x$ and $y$ have the same distance from the top of their respective chains, up to some additive constant. Moreover, the lengths of the two chains differ by at most some additive constant, and the $i^{\text{th}}$ vertex in one chain, when counting from the top, is $O(1)$-close to the $i^{\text{th}}$ vertex in the other chain (if such exists).

We now describe how to map ${\{0,1\}^{n}}$ to ${\mathcal{B}_{n}}$ based on the partition of De Bruijn [et al.]{} Consider a chain $c_k, c_{k+1}, \ldots, c_{n-k}$. Our mapping will “squeeze” the vertices to the top half of the cube 