Geodesic
Two-Point Concentration of the Independence Number of the Random Graph
Forum of Mathematics, Sigma, Tome 12 (2024)

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We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \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 two values for $ p = o ( (\log (n)/n)^{2/3} )$. The extent of concentration of the independence number of $ G_{n,p}$ for $ \omega (1/n) p \le n^{-2/3}$ remains an interesting open question. 
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Tom Bohman; Jakob Hofstad. Two-Point Concentration of the Independence Number of the Random Graph. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.6

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