Two-Point Concentration of the Independence Number of the Random Graph

Abstract

We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ ε} < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on 2 values for $ p = o \left( (\log(n)/n)^{2/3} \right)$. The extent of concentration of the independence number of $ G_{n,p}$ for $ ω(1/n) <p \le n^{-2/3}$ remains an interesting open question.