  For any graph $H$, let ${\rm Forb}^*(H)$ be the class of graphs with no
induced subdivision of $H$. It was conjectured in [A.D. Scott, Induced trees in
graphs of large chromatic number, {\em Journal of Graph Theory}, 24:297--311,
1997] that, for every graph $H$, there is a function $f_H:\mathbb{N}
\rightarrow \mathbb{R}$ such that for every graph $G \in {\rm Forb}^*(H)$,
$\chi(G) \leq f_H(\omega(G))$. We prove this conjecture for several graphs $H$,
namely the paw (a triangle with a pendant edge), the bull (a triangle with two
vertex-disjoint pendant edges), and what we call a "necklace," that is, a graph
obtained from a path by choosing a matching such that no edge of the matching
is incident with an endpoint of the path, and for each edge of the matching,
adding a vertex adjacent to the ends of this edge.

Combinatorics