A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property.
It is well-known that every monotone increasing graph property has a threshold $t(n)$, i.e, $$\lim_{n \rightarrow \infty }P(G_{n,p}~has~property~\mathcal{P})=0$$
$$\lim_{n \rightarrow \infty }P(G_{n,p} \text{ has property }\mathcal{P})=0$$
if $p<<t(n)$$p\ll t(n)$ and $$\lim_{n \rightarrow \infty }P(G_{n,p}~has~property~\mathcal{P})=1$$
$$\lim_{n \rightarrow \infty }P(G_{n,p}\text{ has property }\mathcal{P})=1$$
if $p>>t(n)$$p\gg t(n)$. Here $G_{n,p}$ is a random graph with $n$ vertices where each edge in the graph is added independently with probability $p$.
Property of a graph NOT being planar is monotone increasing. So the first question is what the threshold $t(n)$ for such a property $\mathcal{P}$ is. Does anyone know such a result?
There is another model for random graph, say $G_{n,m}$, which describes a random graph with $n$ vertices and $m$ edges. Similarly we can define a threshold $t^*(n)$ for a given property $\mathcal{P}$ concerning this model, i.e., $$\lim_{n \rightarrow \infty }P(G_{n,m}~has~property~\mathcal{P})=0$$$$\lim_{n \rightarrow \infty }P(G_{n,m}\text{ has property }\mathcal{P})=0$$ if $m<<t^*(n)$$m\ll t^*(n)$ and $$\lim_{n \rightarrow \infty }P(G_{n,m}~has~property~\mathcal{P})=1$$$$\lim_{n \rightarrow \infty }P(G_{n,m}\text{ has property }\mathcal{P})=1$$ if $m>>t^*(n)$$m\gg t^*(n)$.
It is well-known that if $m\geq 3n-5$, then $G_{n,m}$ is not planar and thus $G_{n,m}$ has propery $\mathcal{P}$. However, is there any result for the exact value of $t^*(n)$?
