Determine the sharp constant in the Erdős–Rényi Lovász-number asymptotic

Determine the correct leading constant in the asymptotic behavior of the expected Lovász number for the Erdős–Rényi random graph G(n,p) in the regime \(\log^6 n/n \leq p \leq 1/2\), where the known order is \(\Theta(\sqrt{n/p})\).

Background

The paper reviews prior results on the Lovász number of Erdős–Rényi random graphs. Juhász established that the expected Lovász number has order $\Theta(\sqrt{n/p})$ when $\log^6 n/n \leq p \leq 1/2$, while subsequent work addressed concentration around the median in dense and sparse regimes. Although the order of growth is known, the precise multiplicative constant is unresolved according to the paper.