Tournois infinis et critiques

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for every a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A. For example, ?, {x} (x∈V) and V are intervals of T, called trivial intervals. A tournament all the intervals of which are trivial is called indecomposable; otherwise, it is decomposable. An indecomposable tournament T=(V,A) is then said to be critical if for each x∈V, T(V?{x}) is decomposable and if there are x≠y∈V such that T(V?{x,y}) is indecomposable. We introduce the operation of expansion which allows us to describe a process of construction of critical and infinite tournaments. It follows that, for every critical and infinite tournament T=(V,A), there are x≠y∈V such that T and T(V?{x,y}) are isomorphic. To cite this article: I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 336 (2003).