Let's define the coprime graph $CG(n)$ as a graph, the vertices of which are numbers from $1$ to $n$, two numbers $p$ and $q$ are connected with an edge iff they are coprime. What is the chromatic number of $CG(n)$?
It is clearly not less than the clique number of $CG(n)$, that is equal to the number of primes less than $n$ (which is asymptotically equivalent to $\frac{n}{log(n)}$) and it clearly does not exceed the total number of vertices (which is $n$). However, I would like to see some more accurate asymptotical bounds...
Well, if you already have a color for every prime number, and another color for $1$, which is coprime with everything, then you can just give every composite number the color of one of its prime factors. (Doesn't matter which one!)
Now, if two numbers have the same color, then they share a prime factor, so there's no edge between them.
This uses $\pi(n)+1$ colors, and we already know that's best possible, because there's a clique of order $\pi(n)+1$: all the primes $\le n$, and $1$.