What is the maximum treewidth of a graph with $m$ edges? In other words, what is the correct growth for the following function?
$\alpha(m) = max\{\mathrm{treewidth}(G): G \mbox{ has $m$ edges}\}$.
Clearly $\alpha(m) = \Omega(\sqrt{m})$, which is attained by cliques with $\Omega(\sqrt{m})$ vertices and by by $\Omega(\sqrt{m})\times \Omega(\sqrt{m})$ grids.
Is this also the maximum?