A capacity-achieving sequence of degree distributions for the erasure channel is, roughly speaking, a sequence of degree distributions such that graphs sampled uniformly at random satisfying those degree constraints lead to codes that perform arbitrarily close to the capacity of the erasure channel when decoded with a simple erasure decoder described in the paper. We will prove a necessary property called flatness for a sequence of degree distributions to be capacity-achieving, and will comment on possible applications to the design of capacity-achieving sequences on other communication channels