  We present a natural restriction of Hindman's Finite Sums Theorem that admits
a simple combinatorial proof (one that does not also prove the full Finite Sums
Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet
implies the existence of the Turing Jump, thus realizing the only known lower
bound for the full Finite Sums Theorem. This is the first example of this kind.
In fact we isolate a rich family of similar restrictions of Hindman's Theorem
with analogous properties.
