Let HT denotes the statement of Hindman’s theorem. Within RCA0 one can prove that:
1. HT implies ACA0
2. HT can be proved in ACA0+.
An open question is the strength of Hindman’s theorem.
Is HT equivalent to ACA0+ , or to ACA0, or does it lie strictly between them?
This is a very good question.
A good place to start in answering this question is
https://www.researchgate.net/publication/301854082_Effectiveness_of_Hindman's_theorem_for_bounded_sums
Unfortnately, this paper is not available on RG but it is available a University of Connecticut web page at
http://www.math.uconn.edu/~damir/papers/Hindman.pdf
See also
https://www.researchgate.net/publication/45858014_Hindman's_Theorem_An_Ultrafilter_Argument_in_Second_Order_Arithmetic
where the issue of the strength of Hindman's theorem is raised.
Article Effectiveness of Hindman's theorem for bounded sums
Article Hindman's Theorem: An Ultrafilter Argument in Second Order Arithmetic
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