On the extremal function for graph minors

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