Lovasz number of G(n,1/2)

Establish that for a graph G sampled from G(n, 1/2), the expected Lovasz number satisfies E[ϑ(G)] = (1 + o(1)) √n.

Background

The Lovasz number ϑ(G) is a semidefinite relaxation sandwiched between the clique number of the complement and the chromatic number, and it upper bounds Shannon capacity. Its behavior for Erdős–Rényi G(n,1/2) has been studied, but precise asymptotics are unknown.

The conjectured scaling E ϑ(G) ~ √n mirrors strong pseudorandomness heuristics and is consistent with bounds known in related models.

References

There are several results on the Lovasz number for Erd\H{o}s-R\ enyi random graphs G(n, p) (see). However, the precise asymptotics of the Lovasz number for G(n, 1/2) is still open. Let G be distributed as G(n, 1/2). Then, \begin{equation} \mathbb{E} \vartheta(G) = (1 + o(1))\sqrt{n}. \end{equation}

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Section “The Lovasz number of random circulant graphs (DD)” (Entry 9)