Clique number of Paley graphs is polylogarithmic

Prove that for primes p ≡ 1 (mod 4), the clique number of the Paley graph G_p satisfies ω(G_p) = O(polylog p).

Background

Paley graphs are deterministic pseudorandom graphs expected to mirror properties of G(n,1/2), where clique numbers are logarithmic. Current best upper bounds achieve (1+o(1))√(p/2), far from polylogarithmic.

A polylogarithmic bound would align with pseudorandomness heuristics and transform understanding of extremal subgraph structure in highly algebraic graphs.

References

A random graph of the same degree has logarithmic clique number, which motivates the following well known conjecture (see, e.g., Open Problem 8.4 in) Let $\omega(G_p)$ denote the clique number of the Paley Graph for $p \equiv 1 \pmod{4}$ prime. \ \omega(G_p)=O(\mathrm{polylog(p)}).

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “On the clique number of the Paley Graph (ASB)” (Entry 12)

However, for Paley graphs of nonsquare order, these structural results are not available; in fact, determining the asymptotic of their clique number is a notoriously open problem.

— Positivity preservers over finite fields II  (2608.18978 - Guillot et al., 19 Aug 2026) in Introduction