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, Paley 图表在质数 $ p p $ p 1/ 3} 的 clation 值至少为 $\ Omega (p\ 1/ 3}) 美元。 这与人们普遍认为 Paley 图形的实际 clation 值为 $O (\ mathrm{polylog} (p) 美元。 我们的结果可以被视为 Deshpande 和 Montanari (2015) 的 Oral- caltical $的解析值。 后者显示的值值相同( $\ mathrm{polylog} (p) 3 值) 。 这与人们广泛相信的假设相反, Paley 图形的实际 caltique 值为 $( mathrem) 4( log) 美元。 我们还显示, 我们的下限值是 Feige- Kruppermomental, 但它显示的是 Kelner 的某个参数 $ 。 最后, 我们提出的数字实验显示的是, 该值可能为 4 ligalalalalal lax 。