The Asymptotic χ‐Boundedness of Hereditary Families

A family of graphs is asymptotically ‐bounded with bounding function if almost every graph in the family satisfies . A graph is ‐free if it does not contain as an induced subgraph. We ask which hereditary families are asymptotically ‐bounded, and discuss some related questions. We show that for every tree , almost all ‐free graphs satisfy . We show that for every cycle except , almost every ‐free graph satisfies . We show that the ‐free graphs are asymptotically ‐bounded with bounding function .