We study the conflict-free chromatic number χ CF of graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graph G(n,p) and give the asymptotics for p = ω(1/n). We also show that for p ≥ 1/2 the conflict- free chromatic number differs from the domination number by at most 3.
