  We prove better lower bounds on additive spanners and emulators, which are
lossy compression schemes for undirected graphs, as well as lower bounds on
shortcut sets, which reduce the diameter of directed graphs. We show that any
$O(n)$-size shortcut set cannot bring the diameter below $\Omega(n^{1/6})$, and
that any $O(m)$-size shortcut set cannot bring it below $\Omega(n^{1/11})$.
These improve Hesse's [Hesse03] lower bound of $\Omega(n^{1/17})$. By combining
these constructions with Abboud and Bodwin's [AbboudB17] edge-splitting
technique, we get additive stretch lower bounds of $+\Omega(n^{1/11})$ for
$O(n)$-size spanners and $+\Omega(n^{1/18})$ for $O(n)$-size emulators. These
improve Abboud and Bodwin's $+\Omega(n^{1/22})$ lower bounds.
