Sperner's Lemma can be stated as follows.
Lemma of Sperner. Let $\Omega$ a finite set with $n$ elements ( $|\Omega|=n$). If a family $\{ A_i \}_{1\leq i \leq N}\subset \Omega$ satisfies the Sperner condition, i.e. $A_i\nsubseteq A_j$ and $A_j\nsubseteq A_i$ for all $i \neq j$, then $$ N \leq \frac{n!}{ \left\lfloor \frac{n}{2} \right\rfloor ! \left(n-\left\lfloor \frac{n}{2} \right\rfloor\right) !} $$
Here, $\left\lfloor \frac{n}{2} \right\rfloor$ is the integer part of $ \frac{n}{2}$.
Question. What other statements of Sperner's Lemma exist?
In Wikipedia, for example, there is a different statement of Sperner's Lemma. I ask that your statement of Sperner's Lemma to be accompanied by a proof that the statement of Sperner's Lemma in this question and the statement of Sperner's Lemma in your answer are equivalent or at least an indication of a book, article or link that contains the proof of such equivalence.