Brief introduction on χ-bounded problem

       A family G of graphs are said to be χ-bounded if there exists a function ϕ such that $\chi(G)\le \phi(\omega(G))$ for each graph G of G (Gyarfas 1983). Let F be a family of graphs. A graph G is said to be F -free if G induces no member of F. Erdos's theorem assures that if F-free graphs are χ-bounded, then F is either an infinite family or has to contain a tree. In this talk, we will briefly introduce some new results and still open problems related to χ-bounded problems.