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.
Definitions.
KFL uses the following concepts.
Statement.
Let "T" be a boundary-antipodally-symmetric triangulation of formula_1 and "L" a boundary-odd labeling of "T".
If "L" has no complementary edge, then "L" has an odd number of "n"-dimensional alternating simplices.
Corollary: Tucker's lemma.
By definition, an "n"-dimensional alternating simplex must have labels with "n" + 1 different sizes.
This means that, if the labeling "L" uses only "n" different sizes (i.e. formula_10), it cannot have an "n"-dimensional alternating simplex.
Hence, by KFL, "L" must have a complementary edge.
Proof.
KFL can be proved constructively based on a path-based algorithm. The algorithm it starts at a certain point or edge of the triangulation, then goes from simplex to simplex according to prescribed rules, until it is not possible to proceed any more. It can be proved that the path must end in an alternating simplex.
The proof is by induction on "n".
The basis is formula_11. In this case, formula_1 is the interval formula_13 and its boundary is the set formula_14. The labeling "L" is boundary-odd, so formula_15. Without loss of generality, assume that formula_16 and formula_17. Start at −1 and go right. At some edge "e", the labeling must change from negative to positive. Since "L" has no complementary edges, "e" must have a negative label and a positive label with a different size (e.g. −1 and +2); this means that "e" is a 1-dimensional alternating simplex. Moreover, if at any point the labeling changes again from positive to negative, then this change makes a second alternating simplex, and by the same reasoning as before there must be a third alternating simplex later. Hence, the number of alternating simplices is odd.
The following description illustrates the induction step for formula_18. In this case formula_1 is a disc and its boundary is a circle. The labeling "L" is boundary-odd, so in particular formula_8 for some point "v" on the boundary. Split the boundary circle to two semi-circles and treat each semi-circle as an interval. By the induction basis, this interval must have an alternating simplex, e.g. an edge with labels (+1,−2). Moreover, the number of such edges on both intervals is odd. Using the boundary criterion, on the boundary we have an odd number of edges where the smaller number is positive and the larger negative, and an odd number of edges where the smaller number is negative and the larger positive. We call the former "decreasing", the latter "increasing".
There are two kinds of triangles.
By induction, this proof can be extended to any dimension.