  For any positive integer $t$, a \emph{$t$-broom} is a graph obtained from
$K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for
graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$,
where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of
$G$, respectively. When $t=2$, this answers a question of Schiermeyer and
Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to
$7\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and
\{$t$-broom, $K_{t,t}$\}-free graphs, we improve the bound to $o(\omega^{t})$.
