TITLE: Chromatic number of induced subgraphs as upper bound to the chromatic number
QUESTION [7 upvotes]: Motivation: At the Erdős100 conference in Budapest András Gyárfás presented some interesting conjectures. One of them was the following: 
Given that in a graph $G$, every subgraph $H$ formed by taking the induced subgraph on the vertices of a path, has chromatic number at most $r$. Is it true that then the chromatic number of $G$ is bounded by a function of $r$?
Question: Are there any results such that: If in a graph $G$, every induced subgraph with a special property has chromatic number at most $r$, then the chromatic number of $G$ is bounded by $f(r)$?
Of course i would like to avoid special properties that enable induced subgraphs with a lot of vertices, like the property that "This induced subgraph contains all but one vertex".
EDIT:  Let $G$ be connected.

REPLY [3 votes]: There is a very similar question to that of Gyarfas which is well-known, still open, and notoriously hard.
Question: Is it true that if every triangle-free induced subgraph of $G$ has chromatic number at most $r$, then $G$ has chromatic number at most $f(r,\omega)$, where $\omega$ is the clique number of $G$?
A positive answer to this question would directly imply the following conjecture of Gyarfas:
Conjecture: There is a function $g$ such that every graph with no odd induced cycle of length at least 5, and clique number at most $k$, has chromatic number at most $g(k)$.