Rainbow KKM lemma (Gale)

Statement

David Gale proved the following generalization of the KKM lemma. Suppose that, instead of one KKM covering, we have n different KKM coverings: $C^1_1,\ldots,C^1_n,\ldots,C^n_1,\ldots,C^n_n$. Then, there exists a permutation $\pi$ of the coverings with a non-empty intersection, i.e.:

$$\bigcap_{i=1}^{n}C^{\pi(i)}_i \neq \emptyset.$$

The name "rainbow KKM lemma" is inspired by Gale's description of his lemma:

"A colloquial statement of this result is... if each of three people paint a triangle red, white and blue according to the KKM rules, then there will be a point which is in the red set of one person, the white set of another, the blue of the third".

The rainbow KKM lemma can be proved using a rainbow generalization of Sperner's lemma.

The original KKM lemma follows from the rainbow KKM lemma by simply picking n identical coverings.