Dedicated to Michel Deza at the occasion of his 65 th birthday.Abstract. A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The profile of a tournament T is the function ϕT which counts for each integer n the number ϕT (n) of tournaments induced by T on the n-element subsets of T , isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of ϕT is either polynomial, in which case ϕT (n) ≃ an k , for some positive real a, some non-negative integer k, or as fast as some exponential.

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A binary relation is (≤k)-reconstructible, if it is determined up to isomorphism by its restriction to subsets of at most k elements. In [8], Lopez has shown that finite binary relations are (≤6)-reconstructible. To prove that the value 6 of its result, is optimal, Lopez [3], associates to all finite binary relation, an infinity of finite extensions, that are not (≤5)-reconstructible. These extensions are obtained from the relations given, by creation of intervals. Rosenberg has then asked if all finite binary relations, not (≤5)-reconstructible, were obtained by the same process. In this paper, we give an affirmative answer to the question, by characterizing finite binary relations that are not (≤5)-reconstructible. We deduce the 5-reconstructibility of finite indecomposable binary relations, of at least 9 elements. We extend then this last result to the binary multirelations.