Let $B_n$ be the boolean lattice of rank $n$. Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum, respectively.

We identify the notion of edge with the notion of interval $[a,b]$ of cardinal $2$.

We propose to label every edge with the symbols $\alpha$ or $\beta$, such that:

- For every maximal chain, exactly one edge is labeled by $\alpha$.
- For every element $a \in B_n$, $a \neq \hat{1}$, then at most one edge $[a,x]$ is labeled by $\alpha$.

(note that $x$ is an atom of the interval $[a, \hat{1}]$)

*Question*: Is there a coatom $y$ of $B_n$ such that the edge $[y,\hat{1}]$ is labeled by $\alpha$?