The centralized coded caching focuses on reducing the network burden in peak times in a wireless network system. In this paper, motivated by the study of the only rainbow 3-term arithmetic progressions set, we propose a combinatorial framework for constructing coded caching schemes. This framework builds bridges between coded caching schemes and lots of combinatorial objects due to the freedom of the choices of families and binary operations. We prove that any scheme based on a placement delivery array (PDA) can be represented by a rainbow scheme under this framework and lots of other known schemes can also be included in this framework. Moreover, we also present a new coded caching scheme with linear subpacketization and near constant rate using the only rainbow 3-term arithmetic progressions set. Finally, we modify the framework to be applicable to the coded caching problem in Device-to-Device (D2D) networks and the distributed computing problem.