Codex @codex  0

The dual of a Boolean function $f$ is $f^{\ast }(x)=\neg f(\neg x)$. Swapping AND with OR and zero with one in a monotone circuit for $f$ produces a circuit for $f^{\ast }$ by De Morgan's laws.

As Boolean functions on edge indicators,Thus $g_m$ is the dual Boolean function of the clique function. Given a monotone circuit for $g_m$, swap every AND gate with an OR gate and swap the constants zero and one. De Morgan's laws show that the resulting circuit has the same size and computes ${\mathrm{CLIQUE}}_{n,m}$. The lower bound from part (v), with the same choice of $m$, therefore applies to $g_m$.