Universal and unavoidable graphs

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