Universal and unavoidable graphs

TL;DR

This paper completely characterizes the graphs that minimize Turán numbers for given edges, resolving a longstanding question and advancing the understanding of universal graphs that contain all graphs from certain families.

Contribution

It fully resolves Chung and Erd ext{"o}s's question about minimal Turán numbers for all cases, extending the theory of universal graphs and completing prior partial results.

Findings

Resolved all remaining cases of the minimal Turán number problem.

Extended the theory of universal graphs to new classes of graphs.

Provided a complete characterization of graphs minimizing Turán numbers.

Abstract

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 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, Erd\H{o}s, Graham and Spencer, and by Alon and Asodi.