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We give a lower bound of $\tilde{\Omega}(\sqrt{n})$ for the degree-4Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, weshow that on an Erd\"os-R\'enyi graph $G(n,\tfrac{1}{2})$, with highprobability there is a feasible point for the degree-4 SOS relaxation of theclique problem with an objective value of $\tilde{\Omega}(\sqrt{n})$, so thatthe program cannot distinguish between a random graph and a random graph with aplanted 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 givelower 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 SDPsolution proposed in their work, then showing that this perturbation remainsPSD as the objective value approaches $\tilde{\Omega}(n^{1/2})$.  In an independent work, Hopkins, Kothari and Potechin [HKP15] have obtained asimilar lower bound for the degree-$4$ SOS relaxation.

Tight Lower Bounds for Planted Clique in the Degree-4 SOS Program