  The Gy\'arf\'as-Sumner conjecture says that for every tree $T$ and every
integer $t\ge 1$, if $G$ is a graph with no clique of size $t$ and with
sufficiently large chromatic number, then $G$ contains an induced subgraph
isomorphic to $T$. This remains open, but we prove that under the same
hypotheses, $G$ contains a subgraph $H$ isomorphic to $T$ that is
``path-induced''; that is, for some distinguished vertex~$r$, every path of $H$
with one end $r$ is an induced path of $G$.
