I will discuss a recent work (arxiv:2004.07282) on α' corrections to the 4-pt functions of half-BPS operators in N=4 SYM which are dual to the scattering of four Kaluza-Klein modes on AdS_5 x S^5. Crossing symmetry and a set of constraints on the form of the spectrum fully fix the correlator at the order α'^5. The result exhibits a number of natural patterns which suggest that the bootstrap approach outlined here will extend to higher orders in a simple way.