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  "body": "\n## Statement\n\nDavid 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.:\n\n$$\\bigcap_{i=1}^{n}C^{\\pi(i)}_i \\neq \\emptyset.$$\n\nThe name \"rainbow KKM lemma\" is inspired by Gale's description of his lemma:\n\n\u003e \"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\".\n\nThe rainbow KKM lemma can be proved using a rainbow generalization of Sperner's lemma.\n\nThe original KKM lemma follows from the rainbow KKM lemma by simply picking *n* identical coverings.\n",
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