A chi-bounded graph class is a class of graphs for which a function 
of the clique number bounds
the chromatic number of every induced
subgraph of every graph in the class. Thus 
for each such nonempty induced
subgraph 
, where 
and 
denote its chromatic
number and clique number, respectively. For
a class closed under taking induced subgraphs,
it suffices to check the inequality on the members of the class (Scott and Seymour
2020). The class of perfect graphs is chi-bounded
by 
.
The Gyárfás-Sumner conjecture asserts that for every fixed forest 
, the class of graphs with
no induced subgraph isomorphic
to 
is chi-bounded (Gyárfás 1987, Scott and Seymour 2020). Bierbaumer and
Bucić (2026) proved that for every fixed forest 
and positive
integer 
, such a graph 
on 
vertices with 
satisfies 
for a constant 
. The conjecture asks for a bound independent
of 
.
-Boundedness." J. Graph Th. 95, 473-504,
2020.