of counting poly\] *(i)* we have $${\mathbb E}_{T \sim {\cal T}(n,\alpha )}(X_U) = (|U|)_h \times p_H(\alpha ) = (|U|)_h \times 2^{-\binom {h}{2}}.$$ Since each edge belongs to at most $\binom{h}{2}n^{h-2}$ copies of $H$, $X_U$ is sharply concentrated by Azuma’s inequality (see Chapter 7 [@AS]) $$\mathbb{P}(|X_U-\mathbb{E}(X_U)|>
    \varepsilon n^h) \le 2e^{\frac{-\varepsilon^2n^{2h}}{2\binom{n}{2}\left(\binom{h}{2}n^{h-2}\right)^2}}=e^{-\Omega(n^2)}.$$ In particular, by a union bound with probability $1-o(1)$ we have $X_U = 2^{-\binom {h}{2}}|U|^h + o(n^h)$ for every $U \subset V(T)$.

On the other hand the number of forward edges of $T$ is distributed as $\operatorname{Bin}(\binom{n}{2},\alpha)$ so by standard Chernoff type estimates $T$ has $\alpha \binom {n}{2} \pm o(n^2)$ forward edges with probability $1 - o(1)$ (see Appendix A of [@AS]). Combined with the previous paragraph, we have shown that there is an $n$-vertex tournament $T$ which satisfies ${\cal P}^*_2(H)$ but not ${\cal P}_1$. Thus $H$ cannot be locally forcing.

For the rest of this section it will be more convenient to work with a rescaled and recentered counting polynomial $q_H(x) := 2^{\binom {h}{2}} \times p_H\big ( \frac{1+x}{2} \big )$. We now show several restrictions that the [$\mathcal{T}_{cliq}$-forcing]{} condition imposes on a tournament. The first one is that a [$\mathcal{T}_{cliq}$-forcing]{} tournament should be almost regular.

\[lem: restrict-simple-degree\] If $H$ is 