We study a new bin packing problem with a color constraint. A finite set of items and an unlimited number of identical bins are given. Each item has a set of colors. Each bin has a color capacity. The set of colors for a bin is the union of colors for its items and its cardinality can not exceed the bin capacity. We need to pack all items into the minimal number of bins. For this NP-hard problem, we design the core heuristic based on the column generation approach for the large-scale formulation. A hybrid VNS matheuristic with large neighborhoods is used for solving the pricing problem. We use our core heuristic in the exact branch-and-price method. Computational experiments illustrate the ability of the core heuristic to produce optimal solutions for randomly generated instances with the number of items up to 250. High-quality solutions on difficult instances with regular structure are found.