Graph coloring minimizing the number of colors in every independent set Is the following claim known?

Claim: For any graph $G$ with $n$ vertices there exists a coloring of $G$ such that every independent set is colored by at most $O(\sqrt{n})$ colors.
The following claim is known to me, but may not count because it is unpublished:
Any graph on $n$ vertices can be colored so that any induced subgraph $H$ of chromatic number at most $k$ uses at most $\chi(H)+B$ colors,
where $B(B+1)\leq 2kn$.

This is a proof by induction; the motivation was to consider colorings which use few colors not only on the graph but also on all induced subgraphs.
I am not aware of any published results, though.