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)