Authors’ abstract: We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of S. Klainerman et al. [The global nonlinear stability of the Minkowski space. Princeton, NJ: Princeton University Press (1993; Zbl 0827.53055); The evolution problem in general relativity. Boston, MA: Birkhäuser (2003; Zbl 1010.83004)].