A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

@article{Carlucci2018AWV,
  title={A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem},
  author={Lorenzo Carlucci},
  journal={Archive for Mathematical Logic},
  year={2018},
  volume={57},
  pages={381-389}
}
  • L. Carlucci
  • Published 18 October 2016
  • Mathematics
  • Archive for Mathematical Logic
Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over $$\mathbf {RCA}_0$$RCA0 to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey… 
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Upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem are proved, highlighting the role of a sparsity-like condition on the solution set, which is called apartness.
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New Bounds on the Strength of Some Restrictions of Hindman's Theorem
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