I want to prove the following theorem:
With high probability, every subgraph $G\subset G(n,1/2)$ with $$e(G)\geq \left(\frac{1}{2}+\epsilon\right)\frac{n^2}{4}$$ contains a triangle.
I think this the usual probabilistic method (proving that $\mathbb{P}(G \text{ is triangle-free})\to 0$) works here. However I don't quite understand how to do this since the variable that tends to infinity is not the number of vertices of $G$, but only the number of vertices of $G(n,1/2)$.
Would the expected number of triangles in $G$ be $$\frac{1}{2^3}\binom{v(G)}{3}?$$ It is not obvious to me which properties $G$ inherits of $G(n,1/2)$.
Can anyone help?