Linear-dimension rigidity in the dense regime at p = 1/2

Determine whether there exists a constant ε > 0 such that the Erdős–Rényi random graph G(n, 1/2) is ε n-rigid asymptotically almost surely.

Background

The authors emphasize that the dense regime (p bounded away from 0 and 1) is particularly challenging for their methods. Even establishing any positive linear rigid dimension for G(n,1/2) is currently beyond reach.

This problem seeks a lower bound showing existence of linear-in-n rigidity dimension for G(n,1/2), contrasting with results in sparser regimes where d(n,p) is controlled by either minimum degree or edge count.