[@CPS] that there is a non-transitive globally forcing tournament $H$ on five vertices (see Theorem 1.1 from [@CPS]). It would perhaps be interesting to determine which, if any, of the remaining small tournaments are also globally forcing.

Locally forcing tournaments
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We have seen that having ‘correct count’ of a fixed tournament $H$ is not enough to guarantee quasirandomness, for essentially any non-transitive $H$. This is a fairly common situation when studying quasirandom properties in general and the key insight to understand why came from Simonovits and Sós [@SS1] who observed that quasirandomness is a hereditary property, in the sense that any large subgraph of a random structure must also be random-like.

This leads us to the natural question of whether requiring that $T$ contains the ‘correct count’ of $H$ in all large subsets of $V(T)$ is sufficient to guarantee quasirandomness of $T$. To be more precise, consider the following definition.

Let $H$ be an $h$-vertex tournament. Given an $n$-vertex tournament $T$, consider the following property:

-   ${\cal P}^*_2(H)$: $N^*_T(H;U) = 2^{-\binom {h}{2}}|U|^h \pm o(n^h)$ for every set $U \subset V(T)$.

The tournament $H$ is said to be *locally forcing* if ${\cal P}^*_2(H) \implies {\cal P}_1$.

An analogous property for 