= Poisson threshold for vertices lying in no triangle
{c}
{title2=$\binom{n-1}{2}p^3=\log(n/c)\Longrightarrow T\Rightarrow\operatorname{Po}(c)$}
For a fixed set of $r$ vertices, exclude all triangles meeting that set. Their total expectation is $r\log(n/c)+o(1)$. Distinct triangle pairs sharing an edge contribute $O_r(n^3p^5)=o(1)$ to the <Janson dependency sum>. <Harris' inequality> and <Janson inequality> bound the avoidance probability between matching exponentials, giving $(c/n)^r(1+o(1))$. The <factorial-moment criterion for Poisson convergence> then gives the displayed law.
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