Score vectors of tournaments

Abstract A 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.