 Negative colouring error of a forced-set closure

Independently colour each vertex with one of $m-1$ colours and join differently coloured vertices. This complete multipartite graph has no $m$-clique. If a forced set $W$ is rainbow but its $r$ witnesses are not, condition on the colours of $W$. The disjoint petals make those failure events conditionally independent. For each witness of size at most $\ell$, the probability of a colour collision is at most $\left(\genfrac{}{}{0ex}{}{\ell }{2}\right)/(m-1)$. If this is at most $1/2$, one closure step causes an error with probability at most $2^{-r}$. There are at most $n^{\ell +1}$ possible steps; apply the union bound.