arguments allow us a lot of freedom to modify the above procedure in order to obtain examples for other $h$ as well.

All our results are based on a characterisation of locally forcing tournaments in terms of certain graph polynomials. In order to motivate how these polynomials arise let us describe two natural candidates for tournaments with correct local counts of a tournament $H$, which are not quasirandom.

Given $\alpha \in [0,1]$, let ${\cal T}(n,\alpha )$ denote the distribution on the set of tournaments with vertex set $\{1,\ldots, n\}$ in which each pair $\{i,j\}$ with $1 \leq i < j \leq n$ appears independently as $\overrightarrow {ij}$ with probability $\alpha$. We denote by $\mathcal{T}_{cliq}$ the collection of all distributions ${\cal T}(n,\alpha )$ as $\alpha$ varies.

Loosely speaking we say that a graph $H$ is [$\mathcal{T}_{cliq}$-forcing]{} if no ${\cal T}(n,\alpha )$ shows that $H$ is not locally forcing. In other words if there is no $\alpha \neq 1/2$ such that ${\cal T}(n,\alpha )$, which is clearly not quasirandom, has the same local counts of $H$ as ${\cal T}(n,1/2)$ (which [*is*]{} quasirandom).

Given $\alpha \in [0,1]$, let ${\cal T}(n,n, \alpha)$ denote the distribution on the set of tournaments with vertex set $\{1,\ldots, 