Thin Set Versions of Hindman’s Theorem

We examine the reverse mathematical strength of a variation Hindman’s Theorem (HT) constructed by essentially combining HT with Thin Set to obtain principle that we call thin-HT. This states every coloring c:N→N has an infinite set S⊆N whose finite sums are thin for c, meaning there is i c(s)≠i all nonempty s finitely many distinct elements S. show computable instance thin-HT such solution computes ∅′, as case HT, shown Blass, Hirst, and Simpson (1987). In analyzing this proof, deduce implies ACA0 over RCA0+IΣ20. On other hand, using Rumyantsev Shen’s version Lovász Local Lemma, restriction exactly 2 any diagonally noncomputable degree relative ∅′. Hence no Σ20 solution.