Determine conditions on pn such that the probability that a random graph has at least one triangle goes to zero as n increases

Question

I'm currently struggling with an exercise about random graphs where is requested to determine the conditions on $p_n$ such that the probability that $G(n, p_n)$ has at least one triangle goes to zero as $n → +∞$, also assuming that the probability that three vertices forms a triangle is $p_n^3$. I know I have to apply the first order method but I can't relate the theory with general properties.

Answer 1

Let $T_n$ be the number of triangles in a $G(n,p_n)$ random graph. Then you should be able to get a formula for the expected number of triangles, $ET_n$. In order to ensure no triangles, or usually none, you should make (a) $ET _n$ large, or (b) $ET_n$ small. Can you take it from there?

Answer 2

I don't know much about graph theory but you could work out the probability of that no triangle exists. Starting with one random point A and a random point B, the probability that no triangle exists of the form ABx is given by: $(1-p_n^3)^{n-2}$. Now pick a different point C and the probability that no triangle exists of the form ACx conditional on what we already know is given by $(1-p_n^3)^{n-3}$. Using this strategy you can calculate the probability of A not being in a triangle which will be of the order $(1-p_n^3)^{n^2}$. Can you continue?