Score vectors of tournaments

AbstractA tournament T on any set X is a dyadic relation such that for any x, y ∈ X (a) (x, x) ∉ T and (b) if x ≠ y then (x, y) ∈ T iff (y, x) ∉ T. The score vector of T is the cardinal valued function defined by R(x) = |{y ∈ X : (x, y) ∈ T}|. We present theorems for infinite tournaments analogous to Landau's necessary and sufficient conditions that a vector be the score vector for some finite tournament. Included also is a new proof of Landau's theorem based on a simple application of the “marriage” theorem