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)

Since the summand matrices share the same underlying randomness, it is not clear whether existing black-box results from non-asymptotic free probability---even in light of the recent progress in---apply in our setting. Determining whether those techniques can nevertheless be adapted to recover our results is an interesting open question.

— Sharp Lovasz-Theta Bounds on Random Graphs  (2609.30064 - Potechin et al., 24 Sep 2026) in Section 1, paragraph “Discussion on Graph Matrices and Free Probability”