Ky Fan lemma

In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952.^[1]

In this example, where n = 2, there is no 2-dimensional alternating simplex (since the labels are only 1,2). Hence, there must exist a complementary edge (marked with red).

^ Fan, Ky (1952). "A Generalization of Tucker's Combinatorial Lemma with Topological Applications". The Annals of Mathematics. 56 (3): 431–437. doi:10.2307/1969651. JSTOR 1969651.

[1] Fan, Ky (1952). "A Generalization of Tucker's Combinatorial Lemma with Topological Applications". The Annals of Mathematics. 56 (3): 431–437. doi:10.2307/1969651. JSTOR 1969651.

[1]