Define a graph $G_1$ where the vertices of $G_1$ are the points of the plane $\mathbb{R}^2$, and a pair of vertices $p, q$ is connected by an edge if and only if the Euclidean distance $d(p,q) =1$. The Hadwiger–Nelson unit-coloring problem asks for the "chromatic number of the plane": i.e. what is $\chi( G_1 )$? The best known bounds are that $5 \le \chi( G_1 ) \le 7$, with the lower bound improved from 4 to 5 in a breakthrough result by Aubrey de Grey in 2018.
Apparently Erdős asked about the following variant. Let $S =\{ d_1, d_2, \dots, d_k \}$ be a set of positive numbers. Define $G_S$ with vertices corresponding to points in the plane, and edges corresponding to pairs of points at any distance $d_1$, $d_2$, ..., or $d_k$.
Let $f(k)$ denote the maximum chromatic number $\max \chi(G_S)$ as $S$ ranges over all sets of $k$ positive numbers. Clearly $f(k) \le 7^k$, so $f(k)$ is well defined and is finite for every $k$.
Roughly, how does $f(k)$ grow as $k \to \infty$?
According to Soifer (in The Mathematical Coloring Book), Erdős wrote that "it is not hard to see" that $$ \lim_{k \to \infty} \frac{f(k)}{k} \to \infty.$$ As pointed out in the comments, this follows from the fact that a square grid with $\sqrt{n} \times \sqrt{n}$ points only realizes $O \left( n / \sqrt{\log n} \right)$ distinct distances.
What are the best known upper and lower bounds on $f(k)$?
It was recently shown by Exoo and Ismailescu that $f(2) \ge 6$, but I'm not aware of any nontrivial lower bounds for $f(k)$ even for $k \ge 3$.
Any more information for small $k$ would be great, but I'm especially interested in the rate of growth as $k \to \infty$. Erdős apparently asked whether $f(k)$ grows only polynomially fast, and I am wondering whether this is still an open problem.