Tight asymptotics of clique‐chromatic numbers of dense random graphs

The clique-chromatic number of a graph is the minimum colors required to assign its vertex set so that no inclusion maximal clique monochromatic. McDiarmid, Mitsche, and Prałat proved binomial random G n , 1 2 $G\left(n,\frac{1}{2}\right)$ at most + o ( ) log $\left(\frac{1}{2}+o(1)\right){\mathrm{log}}_{2}n$ with high probability (whp). Alon Krivelevich showed it greater than 2000 $\frac{1}{2000}{\mathrm{log}}_{2}\unicode{x0200A}n$ whp suggested right constant in front logarithm $\frac{1}{2}$ . We prove their conjecture and, beyond that, obtain tight concentration result: χ c = − Θ ln ${\chi }_{c}\left(G\left(n,\frac{1}{2}\right)\right)=\frac{1}{2}{\mathrm{log}}_{2}\unicode{x0200A}n-{\rm{\Theta }}(\mathrm{ln}\unicode{x0200A}\mathrm{ln}\unicode{x0200A}n)$