How complete directed graph with n-vertices is connected to the n-dimensional simplex and its triangulation?
Here is what it boils down to. There are $n$ vertices $v_1,\ldots,v_{n-1},v$. For each $i$, there is an edge either from $v$ to $v_i$ or from $v_i$ to $v$. We need to show that one of the following holds:


There is an edge from $v$ to $v_1$.
There is $i &lt; n-1$ such that there are edges from $v_i$ to $v$ and from $v$ to $v_{i+1}$.
There is an edge from $v_{n-1}$ to $v$.


Let $x_i$ be a Boolean variable denoting that there is an edge from $v$ to $v_i$. Then we need to show that the following DNF is a tautology:
$$
x_1 \lor (\lnot x_1 \land x_2) \lor \cdots \lor (\lnot x_{n-2} \land x_{n-1}) \lor \lnot x_{n-1}.
$$
Hopefully you can manage to prove this yourself. It just states that if we have a string of 0s and 1s, then one of the following must happen:


The string starts with 1.
The string contains 01.
The string ends with 0.


The easiest way to see this is to prove the following:


  If a binary string starts with 0 and ends with 1, it must contain 01.


This is just the trivial one-dimensional case of Sperner's lemma.