On tournament inversion

An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\mathrm{i}\mathrm{n}\mathrm{v}}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$ that result in the transitive tournament. Let ${\mathrm{i}\mathrm{n}\mathrm{v}}_k(n)$ be the maximum of ${\mathrm{i}\mathrm{n}\mathrm{v}}_k(T)$ taken over $n$-vertex tournaments. It is well-known that ${\mathrm{i}\mathrm{n}\mathrm{v}}_2(n)=(1+o(1))n^2\mathrm{/}4$ and it was recently proved by Alon et al. that $\mathrm{i}\mathrm{n}\mathrm{v}(n):={\mathrm{i}\mathrm{n}\mathrm{v}}_n(n)=n(1+o(1))$. In these two extreme cases ($k=2$ and $k=n$), random tournaments are asymptotically extremal objects. It is proved that the random tournament {\em does not} asymptotically attain ${\mathrm{i}\mathrm{n}\mathrm{v}}_k(n)$ when $k\ge k_0$ and conjectured that ${\mathrm{i}\mathrm{n}\mathrm{v}}_3(n)$ is (only) attained by (quasi) random tournaments. It is further proved that $(1+o(1)){\mathrm{i}\mathrm{n}\mathrm{v}}_3(n)\mathrm{/}{n}^2\in [\frac{1}{12},0.0992)$ and $(1+o(1)){\mathrm{i}\mathrm{n}\mathrm{v}}_k(n)\mathrm{/}{n}^2\in [\frac{1}{2k(k-1)}+{\delta }_k,\frac{1}{2\lfloor k^2\mathrm{/}2\rfloor }-{\epsilon }_k]$ where ${\epsilon }_k>0$ for all $k\ge 3$ and ${\delta }_k>0$ for all $k\ge k_0$.