prove that many of our examples of [$\mathcal{T}_{cliq}$-forcing]{} tournaments are in fact locally forcing.

**Notation:** Before closing this section we introduce some further notation. Given $m,n \in {\mathbb N}$, let $[n] = \{1,\ldots ,n\}$ and $(n)_m$ denote the falling factorial $(n)_m = n(n-1)\cdots (n-m+1)$. Given a set $X$ we write $\binom{X}{k}$ for the collection of $k$-element subsets of $X$. Given a probability distribution ${\cal D}$ on $X$ we write $x \sim {\cal D}$ to denote an element $x\in X$ selected randomly according to ${\cal D}$. We simply write $x \sim X$ when ${\cal D}$ is taken to be the uniform distribution on $X$. All our logarithms are natural so in base $e$.

Globally forcing tournaments
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In this section we will give the short proof of Proposition \[prop: forcing iff transitive\]. We first show that asymptotically the number of transitive subtournaments of order $h$ in a tournament of order $n$ is minimised by the random tournament. This is precisely the content of Exercise 10.44(b) of [@lovasz-book]. We need a variation of this proof, which allows us to also show the fact that any asymptotic minimiser must be quasirandom. This implies that $Tr_h$ is globally forcing for $h \ge 4$ showing 