How complete directed graph with n-vertices is connected to the n-dimensional simplex and its triangulation?

Question

Answer https://stackoverflow.com/a/26151549/1375882 suggests that Sperner's lemma can be used to prove the existence of index for the search Hamiltonian path in complete directed graph. But Sperner's lemma is about simplex, its triangulation and colouring. How such notions can be mapped to the complete graph with n-vertices?