The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph G_{n,p} for most edge-probabilities p in the range n^{-2/5} \ll p \ll 1. This resolves open problems and questions of Lichev, Mitsche and Warnke as well as Alon and Krievelevich. One major proof difficulty stems from high-degree vertices, which prevent maximal cliques in their neighborhoods: we deal with these vertices by an intricate union bound argument, that combines the probabilistic method with new degree counting arguments in order to enable Janson's inequality. This way we determine the asymptotics of the clique chromatic number of G_{n,p} in some ranges, and discover a surprising new phenomenon that contradicts earlier predictions for edge-probabilities p close to n^{-2/5}.

翻译：图的团染色数是指一种顶点染色所需的最少颜色数，使得不存在极大团是单色的。本文针对边概率 p 在 n^{-2/5} ≪ p ≪ 1 范围内的大部分取值，确定了随机图 G_{n,p} 的团染色数的数量级。这解决了 Lichev、Mitsche 与 Warnke 以及 Alon 和 Krievelevich 提出的公开问题与疑问。一个主要的证明难点源于高度数顶点，它们阻碍了其邻域中的极大团：我们通过一个复杂的并集界论证来处理这些顶点，该论证结合了概率方法以及新的度数计数论证，从而能够应用 Janson 不等式。通过这种方式，我们在某些范围内确定了 G_{n,p} 的团染色数的渐近性，并发现了一个令人惊讶的新现象，该现象与先前对于边概率 p 接近 n^{-2/5} 时的预测相矛盾。