n 1974, erd\h{o}s and rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, boolean lattices and sets. in this talk, i will first give an overview of some previous results on this topic. the second half of this talk is to explore the rainbow variants of the erd\h{o}s-rothschild problem. with jozsef balogh, we confirm conjectures of benevides, hoppen and sampaio, and hoppen, lefmann, and odermann, and complete the characterization of the extremal graphs for the edge-colorings without rainbow triangles. next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. the latter is joint work with yangyang cheng, yifan jing, wenling zhou and guanghui wang.