families of graphs.

In the rest of this subsection we present our main results, which give a good understanding of locally forcing tournaments and show a surprisingly different behaviour compared to the one conjectured by Simonovits and Sós in the graph case. Our first result shows that in order for a tournament to be locally forcing it must be quite strongly quasirandom in several ways.

\[thm: locally-forcing\] Any non-transitive, locally forcing $h$-vertex tournament $H$ satisfies

(i) $$\sum _{v \in V(H)} \big (d^+_H(v) - d^-_H(v)\big )^2 \leq h(h-1).$$

(ii) $|F_{\sigma, H}|=\frac{1}{2}\binom{h}{2} \pm h^{3/2}\sqrt{\log h}$ for every ordering $\sigma$ of $H$.

(iii) For any disjoint subsets $U,W \subseteq T$ we must have $|e(U,W)|\le \frac{1}{2}|U||W| + 2h^{3/2}$

Given that Theorem \[thm: locally-forcing\] says that any locally forcing graph must be strongly quasirandom a natural guess for an example might be to take the random tournament, obtained by orienting each edge of a complete graph uniformly at random, independently between edges. Our next result shows that quite surprisingly with positive probability the random tournament fails condition (i) of Theorem \[thm: locally-forcing\].

\[cor:random-is-not-locally-forcing\] The random tournament is not locally forcing with positive probability.

This tells us that a positive proportion of all tournaments are in 