160 CHAPTER 6. FIXED POINT THEOREMS

Then repeat the process with ∏i≤k+1 p ji valued simplices on[x j1 , . . . ,x jk+1 , x̂ jk+2

]which

have not been crossed. Repeating the process, entering from the outside, cannot deliver a∏

k+2i=1 p ji valued simplex encountered earlier because of what was just noted. There is either

one or two ways to cross the simplices. In other words, the process is one to one in select-ing a ∏i≤k+1 p ji simplex from crossing such a simplex on the selected face of Ŝ. Continuedoing this, crossing a ∏i≤k+1 p ji simplex on the face of Ŝ which has not been crossed pre-viously. This identifies an odd number of simplices having value ∏

k+2i=1 p ji . These are the

ones which are “accessible” from the outside using this process. If there are any which arenot accessible from outside, applying the same process starting inside one of these, leads toexactly one other inaccessible simplex with value ∏

k+2i=1 p ji . Hence these inaccessible sim-

plices occur in pairs and so there are an odd number of simplices in the triangulation havingvalue ∏

k+2i=1 p ji . We refer to this procedure of labeling as Sperner’s lemma. The system of

labeling is well defined thanks to the assumption that {xk−x0}nk=1 is independent which

implies that {xk−xi}k ̸=i is also linearly independent. Thus there can be no ambiguity inthe labeling of vertices on any “face” the convex hull of some of the original vertices of S.The following is a description of the system of labeling the vertices.

Lemma 6.2.1 Let [x0, · · · ,xn] be an n simplex with {xk−x0}nk=1 independent, and let

the first n+ 1 primes be p0, p1, · · · , pn. Label xk as pk and consider a triangulation ofthis simplex. Labeling the vertices of this triangulation which occur on

[xk1 , · · · ,xks

]with

any of pk1 , · · · , pks , beginning with all 1 simplices[xk1 ,xk2

]and then 2 simplices and so

forth, there are an odd number of simplices[yk1

, · · · ,yks

]of the triangulation contained in[

xk1 , · · · ,xks

]which have value pk1 · · · pks . This for s = 1,2, · · · ,n.

A combinatorial method

We now give a brief discussion of the system of labeling for Sperner’s lemma from thepoint of view of counting numbers of faces rather than obtaining them with an algorithm.Let p0, · · · , pn be the first n+1 prime numbers. All vertices of a simplex S = [x0, · · · ,xn]having {xk−x0}n

k=1 independent will be labeled with one of these primes. In particular,the vertex xk will be labeled as pk. The value of a simplex will be the product of its labels.Triangulate this S. Consider a 1 simplex coming from the original simplex

[xk1 ,xk2

], label

one end as pk1 and the other as pk2 . Then label all other vertices of this triangulation whichoccur on

[xk1 ,xk2

]either pk1 or pk2 . The assumption of linear independence assures that

no other vertex of S can be in[xk1 ,xk2

]so there will be no inconsistency in the labeling.

Then obviously there will be an odd number of simplices in this triangulation having valuepk1 pk2 , that is a pk1 at one end and a pk2 at the other. Suppose that the labeling has beendone for all vertices of the triangulation which are on

[x j1 , . . .x jk+1

],{

x j1 , . . .x jk+1

}⊆ {x0, . . .xn}

any k simplex for k≤ n−1, and there is an odd number of simplices from the triangulationhaving value equal to ∏

k+1i=1 p ji . Consider Ŝ ≡

[x j1 , . . .x jk+1 ,x jk+2

]. Then by induction,

there is an odd number of k simplices on the sth face[x j1 , . . . , x̂ js , · · · ,x jk+1

]having value ∏i̸=s p ji . In particular the face

[x j1 , . . . ,x jk+1 , x̂ jk+2

]has an odd number of

simplices with value ∏k+1i=1 p ji := P̂k. We want to argue that some simplex in the triangu-

lation which is contained in Ŝ has value P̂k+1 := ∏k+2i=1 p ji . Let Q be the number of k+ 1