# On the extremal function for graph minors

## TL;DR

This paper establishes an upper bound on the extremal function for graph minors, relating the minimum edge density to the average degree of the minor, with implications for understanding graph structure.

## Contribution

It provides a new upper bound on the extremal function for graph minors that matches known lower bounds for most graphs, advancing the understanding of graph minor theory.

## Key findings

- Derived an explicit upper bound involving average degree and logarithmic factors.
- Matched the upper bound with existing lower bounds for almost all graphs.
- Enhanced the theoretical understanding of the relationship between edge density and minors.

## Abstract

For a graph $H$, let $c(H)=\inf\{c\,:\,e(G)\geq c|G| \mbox{ implies } G\succ H\,\}$, where $G\succ H$ means that $H$ is a minor of $G$. We show that if $H$ has average degree $d$, then $$ c(H)\le (0.319\ldots+o_d(1))|H|\sqrt{\log d} $$ where $0.319\ldots$ is an explicitly defined constant. This bound matches a corresponding lower bound shown to hold for almost all such $H$ by Norin, Reed, Wood and the first author.