Local lemma Ramsey lower bound ($R(k)\ge (\sqrt{2}/e+o(1))k{2}^{k/2}$)

In an independent random two-colouring of $K_N$, a given $k$-vertex set is monochromatic with probability $2^{1-\left(\genfrac{}{}{0ex}{}{k}{2}\right)}$. Events on sets intersecting in at most one vertex use disjoint edges. Thus dependency degree is at most $\left(\genfrac{}{}{0ex}{}{k}{2}\right)\left(\genfrac{}{}{0ex}{}{N-2}{k-2}\right)$. The Lovász local lemma gives a colouring avoiding all monochromatic $K_k$ when $e{2}^{1-\left(\genfrac{}{}{0ex}{}{k}{2}\right)}(1+(\genfrac{}{}{0ex}{}{k}{2}\left)\right(\genfrac{}{}{0ex}{}{N-2}{k-2}))\le 1$. For $N=ck{2}^{k/2}$ the expression is $O(k^{3/2}(ec/\sqrt{2}{)}^k)$, by Stirling's approximation. Taking $c$ tending sufficiently slowly to $\sqrt{2}/e$ proves $R(k)\ge (\sqrt{2}/e+o(1))k{2}^{k/2}$.