Simple one-point extensions of tournaments

Extract

A tournament is a relational structure on the non-empty set T such that for x, y ∈ T exactly one of the three relations

holds. Here x → y expresses the fact that {x, y} ∈ → and we sometimes write this in the alternative form y ← x. Extending the notation to subsets of T we write A → B or B ← A if a → b holds for all pairs a, b with a ∈ A and b ∈ B. is a subtournament of , and is an extension of , if T′ ⊂ T and →′ is the restriction of → to T′; we will usually write 〈′, → 〉 instead of 〈 ′, → ′〉. In particular, if |T − T′| = k, we call a k-poinf extension of .