Lovasz theta of the 1-localization of Paley graphs

Determine the asymptotic behavior of the Lovasz theta function of the complement of the 1-localization of the Paley graph and show that ϑ(Ḡ_{p,1}) ~ √(p/2) for primes p ≡ 1 (mod 4).

Background

The 1-localization G_{p,1} is the induced subgraph on neighbors of a fixed vertex in the Paley graph. Empirical evidence suggests ϑ(Ḡ_{p,1}) asymptotically matches √(p/2), which would recover the Hanson–Petridis bound via an SDP relaxation (that reduces to an LP by circulant structure).

This would validate localization-based convex relaxations for bounding Paley clique numbers.

References

Conjecture $\vartheta(\overline{G_{p,1}) \sim \sqrt{p/2}$ (for $p \equiv 1 \pmod{4}$ prime) where $G_{p,1}$ is the 1-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)