Threshold for unanimity from a random one-half coloring

Establish whether p=C_ε n^{-1} is sufficient for the initially larger side to achieve ε-unanimity in majority dynamics on G(n,p) when the initial vertex colors are independent and uniformly Red or Blue.

Background

Under the random 1/2 coloring scheme, each vertex is initially colored Red or Blue independently with equal probability. The central threshold question asks for the minimum edge probability p such that the side with the initial majority wins with probability at least 1−ε.

The paper relates this problem to the fixed-advantage model and notes that the connectivity threshold p≈n^{-1}log n is necessary to avoid monochromatic isolated components or “echo chambers.” It also records a conjecture of Benjamini, Chan, O’Donnell, Tamuz, and Tan predicting a substantially sharper n^{-1} threshold for ε-unanimity; the paper’s result gives a lower bound of order n^{-2/3}log^{2/3}n but does not settle that conjecture.