Tight asymptotics of clique‐chromatic numbers of dense random graphs

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