  A double star is a tree with two internal vertices. It is known that the
Gy\'arf\'as-Sumner conjecture holds for double stars, that is, for every double
star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an
induced subgraph then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the
chromatic number and the clique number of $G$). Here we prove that $f$ can be
chosen to be a polynomial.
