A degree 4 sum-of-squares lower bound for the clique number of the Paley graph

TL;DR

This paper establishes a degree 4 sum-of-squares lower bound of at least (p^{1/3}) for the clique number of Paley graphs, contrasting with the conjecture that the clique number is polylogarithmic, and explores implications for SOS relaxations.

Contribution

The paper proves a new lower bound for the SOS relaxation of the Paley graph's clique number and analyzes its optimality and limitations compared to random graphs.

Findings

Degree 4 SOS lower bound is (p^{1/3}) for Paley graphs.

Lower bound is optimal for Feige-Krauthgamer pseudomoments.

Numerical experiments suggest SOS may scale as (p^{1/2 - \u03b5}) but cannot surpass the (p^{1/3}) barrier.

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 $\Omega(p^{1/3})$. 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 $1/3$.