  We give a lower bound of $\tilde{\Omega}(\sqrt{n})$ for the degree-4
Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we
show that on an Erd\"os-R\'enyi graph $G(n,\tfrac{1}{2})$, with high
probability there is a feasible point for the degree-4 SOS relaxation of the
clique problem with an objective value of $\tilde{\Omega}(\sqrt{n})$, so that
the program cannot distinguish between a random graph and a random graph with a
planted clique of size $\tilde{O}(\sqrt{n})$. This bound is tight.
  We build on the works of Deshpande and Montanari and Meka et al., who give
lower bounds of $\tilde{\Omega}(n^{1/3})$ and $\tilde{\Omega}(n^{1/4})$
respectively. We improve on their results by making a perturbation to the SDP
solution proposed in their work, then showing that this perturbation remains
PSD as the objective value approaches $\tilde{\Omega}(n^{1/2})$.
  In an independent work, Hopkins, Kothari and Potechin [HKP15] have obtained a
similar lower bound for the degree-$4$ SOS relaxation.
