= Negative colouring error of a forced-set closure
{title2=$\mathbb P(G_{\mathrm{colour}}\in\delta_\vee)\leq n^{\ell+1}2^{-r}$}
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 $\binom{\ell}{2}/(m-1)$. 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>.
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