Universal and unavoidable graphs

The Turan number ex(n, H) of a graph H is themaximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdos asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdos, Graham and Spencer, and by Alon and Asodi.