Lovasz theta of the 2-localization of Paley graphs

Prove that for primes p ≡ 1 (mod 4) and sufficiently large p, the Lovasz theta function of the complement of the 2-localization of the Paley graph satisfies ϑ(Ḡ_{p,2}) ≤ (2/3) √p.

Background

The 2-localization G_{p,2} induces the subgraph on vertices adjacent to two fixed vertices; unlike the 1-localization, it is not circulant. Empirical observations suggest its clique number behaves like (√(1/2) − ε) √p, hinting that θ-based relaxations could improve known bounds.

Establishing the proposed 2/3 √p upper bound on ϑ(Ḡ_{p,2}) would strengthen SDP-based control of Paley cliques beyond 1-localization.

References

Conjecture For $p \equiv 1 \pmod{4}$ prime and large enough we have $\vartheta(\overline{G_{p,2}) \leq \frac23 \sqrt{p}$ where $G_{p,2}$ is the 2-localization of the Paley Graph described above.

— 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)