from ${\cal P}_2(h)$.

Globally forcing tournaments
----------------------------

Although ${\cal P}_2(h)$ guarantees that $T$ is quasirandom, it is natural to ask whether this can already be deduced from the count of a *single* tournament $H$. That is, does quasirandomness of $T$ already follow if $N^*_T(H) = 2^{-\binom {h}{2}}n^h \pm o(n^h)$ for a [single]{} $h$-vertex tournament $H$?

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

-   ${\cal P}_2(H)$: $N^*_T(H) = 2^{-\binom {h}{2}}n^h \pm o(n^h)$.

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

It follows easily from exercise 10.44(b) of [@lovasz-book] that for $h\geq 4$ each transitive tournament $Tr_h$ is globally forcing. This statement was recently reproved in the language of flag-algebras by Coregliano and Razborov [@CR]. Our first observation is that for $h \geq 7$, the transitive $h$-vertex tournament is the *only* tournament with this property.

\[prop: forcing iff transitive\] Let $H$ be an $h$-vertex tournament with $h\geq 7$. Then $H$ is globally forcing if and only if $H$ is transitive.

We remark that, while the condition that $h\geq 7$ may seem unusual in Proposition \[prop: forcing iff transitive\], it has been proven by Coregliano, Parente and Sato 