As far as I understand my literature, the probability of a random graph being fully connected tends toward one as the number of edges approaches a value of size $O(n \log(n))$

The way I read this, there has to be some function $c:[0,1]\to \mathbb{R}$ so that for a given probability one obtains a constant factor for the $n \log(n)$.

More clearly: $c(p)$ should be the value so that after adding $c(p)n\log(n)$ edges the graph is fully connected with probability higher than $p$.

I'd imagine that tight bounds on / descriptions of $c$ are known, but I haven't found any in my (admittedly cursory) reading of the literature. Where should I look ?

The Strange Logic of Random Graphs. – François G. Dorais♦ May 27 '13 at 17:25