Universal and unavoidable graphs. (15th November 2021) - Bucić, Matija; Draganić, Nemanja; Sudakov, Benny
Abstract: The Turán number ex ( n, H ) of a graph H is the maximal number of edges in an H -free graph on n vertices. In 1983, Chung and Erdős 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, Erdős, Graham and Spencer, and by Alon and Asodi.