The extremal function for structured sparse minors

TL;DR

This paper establishes new bounds on the extremal function for structured sparse minors, improving previous results for certain graph classes and providing tight bounds for specific bipartite graphs.

Contribution

It introduces a new upper bound on the extremal function for structured sparse minors and proves a tight lower bound for most such graphs, with applications to bipartite graphs.

Findings

New upper bound on $c(H)$ for structured sparse minors.

Tight lower bound matching the upper bound for most graphs.

Explicit calculation of $c(K_{ft/ ext{log} t,t})$ for certain parameters.

Abstract

Let $c(H)$ be the smallest value for which $e(G)/|G|\geq c(H)$ implies $H$ is a minor of $G$. We show a new upper bound on $c(H)$, which improves previous bounds for graphs with a vertex partition where some pairs of parts have many more edges than others -- for instance a complete bipartite graph with a small number of edges placed inside one class. We also show a tight matching lower bound for almost all such graphs. We apply these results to show $c(K_{ft/\log t,t}) = (0.638\dotsc+o_{f}(1))t\sqrt{f}$, for $f = o(\log t) = \omega(1)$.