We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $\Omega(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is $O(\mathrm{polylog}(p))$. Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to $\mathrm{polylog}(p)$ terms) with high probability for the Erd\H{o}s-R\'{e}nyi random graph on $p$ vertices, whose clique number is with high probability $O(\log(p))$. We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as $O(p^{1/2 - \epsilon})$ for some $\epsilon > 0$, and give a matrix norm calculation indicating that the pseudocalibration proof strategy for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "$\sqrt{p}$ barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from $1/2$ to $1/3$.

翻译：我们证明了素数$p$个顶点上的Paley图的团数的四次平方和（SOS）松弛值至少为$\Omega(p^{1/3})$。这与广泛认为的Paley图实际团数为$O(\mathrm{polylog}(p))$的猜想形成对比。我们的结果可被视为对Deshpande和Montanari（2015）工作的去随机化，他们证明了在$p$个顶点的Erdős–Rényi随机图上，以高概率具有相同的下界（相差$\mathrm{polylog}(p)$因子），而该随机图的团数以高概率为$O(\log(p))$。我们还证明了对于Feige-Krauthgamer的伪矩构造，我们的下界是最优的，这去随机化了Kelner的一个论证。最后，我们展示了数值实验，表明Paley图的四次SOS松弛值可能以$O(p^{1/2 - \epsilon})$（其中$\epsilon > 0$）的规模增长，并给出一个矩阵范数计算，表明用于随机图SOS下界的伪校准证明策略不能直接迁移到Paley图。综合来看，我们的结果表明四次SOS可能打破Paley图团数上界的"$\sqrt{p}$障碍"，但证明它最多能将指数从$1/2$改进到$1/3$。