The asymptotic $\chi$-boundedness of hereditary families

A family ${\cal F}$ of graphs is asymptotically $\chi$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $\chi(G) \le f(\omega(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $\chi$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $\chi(G)=\omega(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $\chi(G) = \omega(G)$. We show that the $C_6$-free graphs are asymptotically $\chi$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$.