Inverting the Tur\'an Problem

TL;DR

This paper explores the inverse of classical extremal graph theory questions, analyzing how the maximum number of edges relates to extremal functions across different graph families and structures.

Contribution

It introduces the concept of inverting extremal problems, showing that standard sequences maximize edges for some graph families but not others, and extends to multigraphs and hypergraphs.

Findings

Standard sequences maximize edges for certain families

Inversion reveals new extremal questions

Extends to multigraphs and hypergraphs

Abstract

Classical questions in extremal graph theory concern the asymptotics of $\operatorname{ex}(G, \mathcal{H})$ where $\mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard' increasing sequence of host graphs $(G_1, G_2, \dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $\operatorname{ex}(G,\mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $\mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.