Exact asymptotic constant for the Lovász number of dense Erdős–Rényi graphs

Determine the correct leading constant in the asymptotic formula for the expected Lovász number of an Erdős–Rényi random graph in the regime where the edge probability satisfies logarithmic-density assumptions and [1m[0mE[0m[1m[0m θ(G(n,p))=[1m[0mΘ[0m[1m[0m(√(n/p)).

Background

The paper reviews known results showing that the expected Lovász number of an Erdős–Rényi random graph is of order √(n/p) in a broad regime of edge probabilities. Although the order of growth is established, the leading constant is not determined. This unresolved issue is presented as a related open problem rather than as a result addressed by the paper, which studies dense random circulant graphs.

References

To the best of our knowledge, determining the correct constant in the Θ(√{n/p}) asymptotic remains an open question.

— The Lovász number of random circulant graphs  (2502.16227 - Bandeira et al., 22 Feb 2025) in Section 1, Introduction

For G(n,\tfrac12), Proposition~\ref{prop:magnitude} confines the constant to [\tfrac12,1]. Which endpoint is correct?

— Induced Embeddings of Graphs into Abelian Cayley Graphs  (2609.01486 - Souop et al., 1 Sep 2026) in Problem 4 (The constant), Section 10 (Open problems)