A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph

Authors Dmitriy Kunisky, Xifan Yu

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Dmitriy Kunisky
  • Department of Computer Science, Yale University, New Haven, CT, USA
Xifan Yu
  • Department of Computer Science, Yale University, New Haven, CT, USA


We thank Afonso Bandeira, Chris Jones, and Daniel Spielman for helpful discussions, and the anonymous reviewers for their careful reading of the paper.

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Dmitriy Kunisky and Xifan Yu. A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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(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 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 construction 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 "√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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Semidefinite programming
  • Mathematics of computing → Combinatorial optimization
  • convex optimization
  • sum of squares
  • Paley graph
  • derandomization


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