Tight Bounds for Hypercube Minor-Universality

Benjamini, Kalifa and Tzalik recently proved that there is an absolute constant $c>0$ such that any graph with at most $c\cdot 2^d\mathrm{/}d$ edges and no isolated vertices is a minor of the $d$-dimensional hypercube $Q_d$, while there is an absolute constant $K>0$ such that $Q_d$ is not $(K\cdot 2^d\mathrm{/}\sqrt{d})$-minor-universal. We show that $Q_d$ does not contain 3-uniform expander graphs with $C\cdot 2^d\mathrm{/}d$ edges as minors. This matches the lower bound up to a constant factor and answers one of their questions.