Complete graph minor density threshold ($c(t)$)

The complete graph minor density threshold isThe density here is edge count divided by order, half the average degree of a vertex. For large $t$, this threshold has order $t\sqrt{\log t}$, with the logarithm taken to base $e$.

For each positive integer $k$, a nonempty graph satisfying $e(G)\ge 11k|G|$ has a graph minor $H$ satisfyingThe density hypothesis is a lower bound. Reversing its inequality would be false for an edgeless graph. This auxiliary lemma can be used to establish a $7t\sqrt{\log t}$ upper bound for the complete graph minor density threshold.