A degree 4 sum-of-squares lower bound for the clique number of the Paley graph
Dmitriy Kunisky, Xifan Yu
TL;DR
This work proves a degree-4 sum-of-squares lower bound for the clique number of Paley graphs, showing $\mathrm{SOS}_4(G_p) \ge c p^{1/3}$ for primes $p \equiv 1 \pmod{4}$. The authors derive this via Feige–Krauthgamer pseudomoments and a careful Schur-complement analysis that leverages Paley symmetry, field- and character-sum estimates, and a graph-matrix decomposition into ribbons and shapes. They also establish that the FK pseudomoments bound is optimal within this restricted framework ($\mathrm{FK}_4(G_p) = \Theta(p^{1/3})$) and provide numerical experiments suggesting a possible $p^{1/2 - \varepsilon}$ scaling for the full SOS$\_4$ relaxation, though the current FK-based construction cannot reach that barrier. Additionally, the paper discusses derandomization implications, showing that while degree-4 SOS may beat the naive $\sqrt{p}$ barrier for Paley graphs upper bounds, the derandomization via graph-matrix norms has limitations, as evidenced by a concrete Paley-specific counterexample. Overall, the results illuminate both the power and limits of SOS methods on pseudorandom deterministic graphs and contribute a rigorous derandomization of prior random-graph lower-bound techniques.
Abstract
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 $Ω(p^{1/3})$. 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 Erdős-Rényi 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 $O(p^{1/2 - ε})$ for some $ε> 0$, 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 $1/3$.