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

We show that the independence number of $G_{n,p}$ is concentrated on two values if $n^{-2\mathrm{/}3+\epsilon }<p\le 1$. 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 $p=o((\log (n)\mathrm{/}n{)}^{2\mathrm{/}3})$. The extent of concentration of the independence number of $G_{n,p}$ for $\omega (1\mathrm{/}n)<p\le n^{-2\mathrm{/}3}$ remains an interesting open question.