Optimal logarithmic factor for complete minors in small-set expanders

Establish whether every $(\alpha,t)$-expander on n vertices contains a complete minor of order at least a constant multiple of $\sqrt{nt/\log t}$, improving the bound $\sqrt{nt/\log n}$ in Theorem 1.3.

Background

The paper proves that, for fixed $\alpha$ and sufficiently large parameters, an $(\alpha,t)$-expander contains a complete minor of order $\Theta_\alpha(\sqrt{nt/\log n})$. The authors note that analogous results for graphs with strong edge-expansion properties achieve a denominator involving $\log d$, where d is the average degree. They leave unresolved whether the small-set-expansion hypothesis alone suffices to replace $\log n$ by $\log t$, which would yield a stronger bound without imposing any assumption on the average degree.