the ‘if’ part of the statement of Proposition \[prop: forcing iff transitive\].

We will first prove that $N^*_T(h):= N^*_T(Tr_h)$ satisfies $N^*_T(h) \geq f_h(n)=(2^{-\binom {h}{2}}-o(1) \big ) { n^h }$ for all $n$-vertex tournaments $T$ where $$f_h(n)=
        \begin{cases}
        \prod_{j=0}^{h-1}\left(\frac{n+1}{2^j}-1\right) \quad &\text{ if } n\ge 2^{h-1}-1\\
        0 \quad &\text{ else.}
        \end{cases}$$

We will prove this by induction on $h$. For $h=1$ and $2$ the result is immediate. Furthermore, $$\label{equation: special case k=3}
            N^*_T(3) 
                    = 
            \sum _{v \in V(T)} \binom {d^+(v)}{2} 
                    \geq
            n \binom {\frac {n-1}{2}}{2} = n(n-1)(n-3)/8=f_3(n).$$ Now assume $h\geq 4$ and $n \ge 2^{h-1}-1$ as otherwise the result is trivial. For each edge $e = \overrightarrow{uv} \in E(T)$ let $N(e) = \{w \in V(T): \overrightarrow{uw}, \overrightarrow{vw} \in E(T)\}$. Clearly $$\label{equation: directed triangle count}
            N^*_T(3) = \sum _{e \in E(T)} 
            |N(e)|.$$ Letting $T[N(e)]$ be the subtournament of $T$ with vertex set $N(e)$, by induction on $h$ $$N^*_T(h) = 
            \sum _{e\in E(T)} 
            N^*_{T[N(e)]}({h-2})
                \geq 
            \sum _{e \in E(T)} 
             f_{h-2}(|N(e)|).$$ It is easy to see that $f_h$ is convex so Jensen’s inequality together with implies $$\begin{aligned}
            \label{equation: transitive subtournaments lower 
            bound}
                N^*_T(h) 
            \geq 
                \binom {n}{2} 
                f_{h-2}{\Bigg ( 
                \frac {N^*_T(3)}{\binom {n}{2}} \Bigg )}
            & = 
                n \cdot \frac{n-1}{2} \cdot f_{h-2}\left(\frac{n-3}{4}\right)
                \nonumber \\ 
            & = 
                n 