Effectiveness Of Hindman'S Theorem For Bounded Sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let H-k(<= n) denote the assertion that for each k-coloring c of N there is an infinite set X subset of N such that all sums Sigma(x is an element of F) x for F subset of X and 0 < vertical bar F vertical bar <= n have the same color. We prove that there is a computable 2-coloring c of N such that there is no infinite computable set X such that all nonempty sums of at most 2 elements of X have the same color. It follows that HT2<2 is not provable in RCA(0) and in fact we show that it implies SRT22 in RCA(0) + BII10. We also show that there is a computable instance of HT3 <= 3 with all solutions computing 0'. The proof of this result shows that HT3 <= 3 implies ACA(0) in RCA(0).