In this talk I will discuss how to construct all solutions consistent with crossing symmetry in the limit of large central charge c ~ N^2, starting from the four-point correlator of the stress tensor multiplet in N=4 SYM. Unitarity forces the introduction of a scale \Delta_{gap} and these solutions organize as a double expansion in 1/c and 1/\Delta_{gap}. These solutions are valid to leading order in 1/c and to all orders in 1/\Delta_{gap} and reproduce, in particular, instanton corrections previously found. Comparison with such instanton computations allows to fix \Delta_{gap}. Using this gap scale one can explain the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge.