The homotopy classes of continuous maps between some non-metrizable   manifolds
  Let R be Alexandroff's long ray. We prove that the homotopy classes of continuous maps R^n \to R are in bijection with the antichains of P({1,...,n}). The proof uses partition properties of continuous maps R^n \to R. We also provide a description of $[X,R]$ for some other non-metrizable manifolds X. 
Mathieu Baillif