On Tournament Inversion

An inversion of a tournament (Formula presented.) is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let (Formula presented.) be the minimum length of a sequence of inversions using sets of size at most (Formula presented.) that result in the transitive tournament. Let (Formula presented.) be the maximum of (Formula presented.) taken over (Formula presented.) -vertex tournaments. It is well known that (Formula presented.) and it was recently proved by Alon et al. that (Formula presented.). In these two extreme cases ((Formula presented.) and (Formula presented.)), random tournaments are extremal objects. It is proved that (Formula presented.) is not attained by random tournaments when (Formula presented.) and conjectured that (Formula presented.) is (only) attained by (quasi)random tournaments. It is further proved that (Formula presented.) and (Formula presented.), where (Formula presented.) for all (Formula presented.) and (Formula presented.) for all (Formula presented.).