This paper investigates the formation composition and keeping problem for multi-quadrotor systems under time-varying communication delays. First, a distributed model predictive control (MPC) approach is employed to transform the formation composition and keeping of multi-quadrotor systems into an online rolling optimization issue, accompanied by the event-triggered mechanism to reduce solution frequency and communication load. Second, to address the asynchrony and time-varying communication delays introduced by event-triggered distributed MPC, a set of constraints is designed to restrict deviations between the current predictive state and previously broadcasted states. Consequently, based on the previously predicted states of neighbors, the cooperation of the multi-quadrotor systems is achieved under asynchronous communication and time-varying delays. This approach guarantees robust asymptotic stability and satisfactory formation performance for multi-quadrotor systems under various delay scenarios. Finally, the numerical and software-in-the-loop (SIL) simulations validate the effectiveness of the proposed algorithm of multi-quadrotors under communication delays.