Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$.


It is well known that $x$ tends to $1$ as $n\to \infty$, even if $p\geq \frac {1+\epsilon}{n}$. However, I am looking for a more exact statement: Something of the form
$$x\geq 1-O(f(n,p))$$
where $f$ is some small function of $n$ and $p$.


I'm sure that a reference exists somewhere, but I have yet to find it.


Thanks!



Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$
In ["On tree census and the giant component in sparse random graphs](http://onlinelibrary.wiley.com/doi/10.1002/rsa.3240010306/abstract), B. Pittel proves the following (see Gap Theorem and Lemma 4): for $c>1,$ any $a>0$ and any $\omega \to \infty$ however slowly, 
\begin{equation\*}
P(|\# \text{ of vertices in largest component}-\theta n |> \omega \sqrt{ n \ln n})\leq n^{-a},
\end{equation\*}
where $\theta$ is the unique positive root of $1-\theta = e^{-c \theta}$. So definitely you can choose $f(n,p)=n^{-a}$ for whichever $a>0$ you want. I imagine that one can mimic the proof to get a better bound when $np \to \infty$. 


