If one plotted the chromatic number $\chi( G_n(p) )$ of an Erdős-Rényi random graph $G_n(p)=G(n,p)$, for a fixed $n$, with respect to probability $p$, what would it look like? For sufficiently large $n$, is there a significant change from low $\chi$ to large $\chi$ at some transition probability $p^*$? And if so, where is that transition probability $p^*$, as a function of $n$?
I assume this is all known, or at least well-studied ...
For convenience, define $b = \frac1{1-p}$. Then we have $$\chi(G_{n,p}) \sim \frac{n}{2 \log_b np}$$ with high probability, which is shown for constant $p$ in (Bollobás, 1988) and for $p \gg \frac1n$ in (Łuczak, 1991).
To give some intuition for this value, the denominator $2 \log_b np$ is the asymptotic value of the independence number $\alpha(G_{n,p})$. So in the random graph, we can take almost all color classes to be almost as large as the maximum possible color class.
The second of these papers also proves bounds on $\chi(G_{n,p})$ when $p = \frac dn$, but we now know much more about that case from (Achlioptas and Naor, 2005). Here, we know that $\chi(G_{n,p})$ is one of two values $\{k_d, k_{d}+1\}$ with high probability, where $k_d$ is the least integer $k$ satisfying $d < 2k\log k$.
(The chromatic number sort of shuffles probability mass from one of these values to the other sufficiently slowly that you "can catch it in the middle" when it can be either value with positive probability; for many values of $d$, we can actually say that it's one or the other.)