Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, April 02, 2020, 12:15 pm

Duration: 30 minutes

Location: Zoom: https://ethz.zoom.us/j/696511724

Speaker: Nemanja Draganic

Universal and unavoidable graphs

The Tur\'an number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erd\H{o}s asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolve this question asymptotically for most of the range of $e$ and ask 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, Erd\H{o}s, Graham and Spencer, and by Alon and Asodi.

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